FORECASTING  THE  YIELD  AND  THE  PRICE 
OF  COTTON 


THE  MACMILLAN  COMPANY 

HEW   YORK   •    BOSTON   •    CHICAGO    -   DALLAS 
ATLANTA    •    SAN    FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON    •    BOMBAY    •  CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


FORECASTING   THE   YIELD   AND 
THE  PRICE  OF  COTTON 


BY 
HENRY  LUDWELL  MOORE 

PROFESSOR  OF  POLITICAL  ECONOMY  IN  COLUMBIA  UNIVERSITY 

AUTHOR  OF  "ECONOMIC  CYCLES:  THEIR  LAW  AND 

CAUSE,"  AND  OF  "  LAWS  OF  WAGES  " 


"  We  have  to  contemplate  social  phenomena  as 
susceptible  of  prevision,  like  all  other  classes, 
within  the  limits  of  exactness  compatible  with 
their  higher  complexity." 

AUGUSTE    COMTE. 


N^m  fork 
THE  MACMILLAN   COMPANY 
1917 

All  rights  reierved 


COPTBIQHT,  1917 

By  the  MACMILLAN  COMPANY 
Setup  and  printed.     Published  October,  1917. 


CD 


?07H 


CONTENTS 


_  CHAPTER  I 

O  PAGE 

Introduction 1 


CHAPTER  II 

J^  THE    MATHEMATICS   OF   CORRELATION 

^      A  Frequency  Distribution 17 

g.      The  Standard  Deviation  as  a  Measure  of  Dispersion       .       .  20 

E      The  Fitting  of  Straight  Lines  to  Data 28 

The  Coefficient  of  Correlation 40 

CHAPTER  III 

THE  GOVERNMENT  CROP  REPORTS 
V) 

hi     The  Character  and  the  Aim  of  the  Crop-Reporting  Service      52 

^     Technical  Terms:  Normal,  Condition,  Indicated  Yield  per 

^  Acre  58 

o     The  Accuracy  of  Forecasts  Tested 65 

ca 

1^     Acreage  and  Production 82 

CHAPTER  IV 

FORECASTING   THE   YIELD   OF   COTTON   FROM   WEATHER  REPORTS 

The  Official  Forecasts  of  the  Yield  of  Representative  States      94 
Forecasting  the  Yield  of  Cotton  from  the  Accumulated  Effects 

of  the  Weather 100 

The  Results  Compared  for  the  Representative  States    .       .115 
Three  Possible  Objections 121 


319341 


vi  Contents 

CHAPTER  V 

THE    LAW   OF   DEMAND   FOR   COTTON 

PAGE 

Two  Practical  Methods  of  Approach 140 

Statics  and  Dynamics  Discriminated 147 

A  Complete  Solution  of  the  Problem 151 

CHAPTER  VI 
Conclusions 163 


FORECASTING  THE  YIELD  AND  THE  PRICE 
OF  COTTON 


FORECASTING  THP:  YIELD  AND 
THE  PRICE  OF  COTTON 

CHAPTER  I 

INTRODUCTION 

An  eminent  economist  has  recently  told  us  that 
economists  no  longer  talk  so  confidently  as  they  once 
did  of  forecasting  social  phenomena,  and  that,  con- 
fronted with  the  complexity  of  social  relations,  'Hhe 
sober-minded  investigator  will  be  slow  in  laying  too 
much  stress  on  single  causes,  slow  in  generalization, 
slowest  of  all  in  prediction."  An  equally  distinguished 
statistician  has  warned  his  colleagues  of  the  dangers 
of  using  refined  mathematical  methods  in  the  treat- 
ment of  the  loose  data  supplied  by  our  official  bureaus. 
These  are  authoritative  warnings,  and  I  have  not  been 
unmindful  of  them  as  the  successive  theses  of  this 
Essay  have  been  developed.  But  the  ultimate  aim  of 
all  science  is  prediction;  the  most  ample  and  trust- 
worthy data  of  economic  science  are  official  statistics; 
and  the  only  adequate  means  of  exploiting  raw^  statis- 
tics are  mathematical  methods. 

The  statistical  devices  used  in  the  treatment  of  our 
problem  of  forecasting  prices  were,  for  the  most  part, 
invented,  for  another  purpose,  by  Professor  Karl  Pear- 
son, and  rest  upon  the  theory  of  probabilities.     Of  the 


2         Forecasting  the  Yield  and  the  Price  of  Cotton 

Pearsonian  superstructure  one  may  repeat  what  La- 
place has  said  of  its  foundation:  '^que  la  theorie  des 
probabilites  n'est,  au  fond,  que  le  hon  sens  reduit  au 
calcul;  elle  fait  apprecier  avec  exactitude  ce  que  les  es- 
prits  justes  sentent  par  une  sorte  d'instinct,  sans  qu'ils 
puissent  souvent  s'en  rendre  compte."  On  the  Cotton 
Exchanges  of  the  world  there  are  always  certain  specu- 
lators, les  esprits  justes  of  the  commodity  market,  who 
seem  to  know  by  a  kind  of  instinct  the  degrees  of  sig- 
nificance to  attach  to  Government  crop  reports,  weather 
reports,  changes  in  supply  and  demand,  and  the  move- 
ments of  general  prices.  Mathematical  methods  of 
probabiHty  reduce  to  system  the  extraction  of  truth 
contained  in  official  statistics  and  enable  the  informed 
trader  to  compute,  with  relative  exactitude,  the  in- 
fluence upon  prices  of  routine  market  factors. 

The  Department  of  Agriculture  of  the  United  States, 
referring  to  the  use  of  its  crop-reporting  service,  has 
briefly  described  the  aim  of  its  admirable  statistical 
organization : 

"Everything  .  ,  .  which  tends  toward  certainty, 
as  regards  either  supply  or  demand,  is  distinctly  ad- 
vantageous to  the  farmer.  Hence  to  throw  light  on 
future  conditions  and  do  away,  as  far  as  possible,  with 
uncertainties  as  to  supply  and  demand,  is  the  principal 
object  of  the  statistical  work  of  the  Department  of 
Agriculture  and  constitutes  the  sole  reason  for  the  col- 
lection of  data  and  the  publication  of  information  re- 
garding current  accumulations  of  farm  products  and 
concerning  crop  conditions  and  prospects.  In  so  far, 
therefore,  as  these  data  are  accurate  and  reliable  — 


Introduction  3 

qualities  which  depend  on  the  integrity  and  intelhgence 
of  crop  correspondents  and  their  interest  in  the  work 
—  the  pubHcation  of  the  information  secured  can  not 
fail  to  reduce  the  uncertainty  regarding  the  future 
values  of  farm  products,  and  thus  have  an  important 
cash  value  to  all  farmers." 

Without  a  doubt  great  values  are  at  stake.  If  the 
size  of  the  cotton  crop  of  1914  is  taken  as  a  standard, 
an  error  in  an  official  crop  report  which  should  lead  to 
an  ultimate  depression  of  one  cent  a  pound  in  the  price 
of  cotton  lint  would  cost  the  farmers  $80,000,000,  or 
more.  A  corresponding  error  leading  to  a  similar  rise 
in  price  would  entail  upon  manufacturers  and  consumers 
a  comparably  heavy  loss. 

The  Department  of  Agriculture  has  rendered  its 
reports  continuously  for  some  fifty  years,  and  yet, 
as  far  as  I  am  aware,  no  one  has  either  measured  the 
degree  of  accuracy  of  the  information  it  supplies  "con- 
cerning crop  conditions  and  prospects,"  or  attempted 
to  see  whether,  by  different  methods,  more  truth  might 
not  be  gained  from  the  stores  of  raw  figures  that  its 
Bureaus  collect. 

Government  Departments  seeking  appropriations 
are  very  likely,  out  of  administrative  necessity,  to 
stress  their  successes  and  suppress  their  failures.  In 
the  January  number  of  the  official  Crop  Reporter,  for 
1900,  this  illustration  is  given  of  the  value  to  planters 
of  the  Government  crop-reporting  service: 

''The  past  year  has  afforded  a  striking  example  of 
the  influence  of  the  reports  on  prices.  As  early  as  Au- 
gust the  Division  of  Statistics  called  attention  to  the 
prevailing  drought  and  its  deleterious  effect  upon  the 


4         Forecasting  the  Yield  and  the  Price  of  Cotton 

growing  crops.  July  1  the  average  condition  reported 
was  87.8;  August  1  it  was  84;  September  1  it  was  68.5; 
October  1,  62.4;  resulting  in  an  average  estimated  yield 
of  lint  cotton  per  acre  of  184  pounds.  Now,  the  lowest 
price  of  futures  in  the  New  York  Cotton  Exchange 
during  1899  was  reached  June  29  when  July  deliveries 
sold  at  5.43.  The  highest  price  was  November  9,  when 
July  deliveries  sold  at  7.74.  Commercial  authorities 
of  high  standing  had  strongly  disputed  the  position 
taken  by  the  Division  of  Statistics,  their  estimates 
running  as  high  in  some  cases  as  12,000,000  bales.  To- 
day the  Department  estimate  of  December  10  of 
8,900,000  bales  is  generally  conceded  to  be  very  close 
to  the  truth,  even  by  these  same  commercial  authori- 
ties. While,  therefore,  the  effect  of  these  overestimates 
was  only  temporary,  it  was,  nevertheless,  sufficient  to 
cause  a  loss  of  several  millions  to  the  cotton  planters." 

This  is  a  success  defiantly  stressed.  We  shall  have 
to  take  but  a  step  to  come  upon  a  failure  ingloriously 
suppressed.  The  reference  to  the  preceding  instance 
is  given  in  the  January  number  of  the  Crop  Reporter, 
for  1900,  p.  2.  But  in  this  same  year  1900,  the  Crop 
Reporter  for  July,  p.  2,  gives  the  following  account  of 
the  condition  eind  prospects  of  the  cotton  crop  for  the 
current  year: 

''Not  only  was  the  condition  on  July  1  for  the  cotton 
region  as  a  whole  the  lowest  July  condition  on  record, 
but  in  Georgia,  Florida,  Alabama,  and  Mississippi 
also  it  was  the  lowest  in  the  entire  period  of  34  years 
for  which  records  are  available,  while  in  Tennessee 
it  was  the  lowest  with  one  exception  and  in  South  Caro- 
lina, Texas,  and  Arkansas  the  lowest  with  two  excep- 


Introduction  5 

tions  in  the  same  period  of  34  years.  Excessive  rains, 
drowning  out  the  crop,  and  followed  by  an  extraordinary 
growth  of  grass  and  weeds,  are  reported  for  almost  every 
State,  and  the  gravity  of  the  situation  is  greatly  in- 
creased by  the  general  scarcity  of  labor.  In  South 
Carolina,  Georgia,  Alabama,  Louisiana,  and  Texas 
considerable  areas  will  have  to  be  abandoned." 

Notwithstanding  this  ill-boding  forecast,  the  rec- 
ords of  the  Bureau  of  Statistics  show  that  the  yield 
per  acre  for  1900  was,  with  the  exception  of  two  years, 
the  largest  in  three  decades.  Later  on,  as  the  crop 
approached  maturity,  the  successive  monthly  reports 
departed  more  and  more  from  the  early  forecast  and 
then  the  official  Bureau  issued  a  final  estimate  that 
approximated  the  truth.  Wlien,  however.  Secretary 
Wilson  of  the  Department  of  Agriculture  was  seeking 
with  the  help  of  Senator  W.  B.  Allison  to  prevent  the 
dupUcation  by  the  Census  Bureau  of  work  usually 
done  by  his  Department,  he  refers  to  the  final  estimate, 
by  his  Department,  of  the  cotton  crop  of  this  same  year, 
1900-1901,  as  ''an  estimate  so  accurate  that  its  sub- 
sequently ascertained  close  agreement  with  actual  pro- 
duction was  commented  upon  throughout  the  entire 
cotton  world  as  a  marvel  of  statistical  forecasting." 
{Crop  Reporter,  March,  1902,  p.  4.) 

Now,  obviously,  what  is  needed  for  business  and  for 
scientific  purposes  is  not  one  or  more  illustrations  either 
in  praise  or  in  blame  of  the  Government  crop-reporting 
service,  but  a  quantitative  testing  of  the  accuracy  of 
the  continuous  service  throughout  a  long  period,  say 
a  quarter  of  a  century.  For  business  and  for  scientific 
purposes  one  must  know  the  degree  of  accuracy  with 


6         Forecasting  the  Yield  and  the  Price  of  Cotton 

which,  upon  an  average,  from  the  official  data  avail- 
able at  any  time,  one  can  forecast  the  ultimate  yield. 

On  many,  if  not  upon  all  the  Cotton  Exchanges  of 
the  country,  the  daily  variations  of  rainfall  and  temper- 
ature in  the  states  of  the  Cotton  Belt,  during  the  grow- 
ing season,  are,  for  the  information  of  brokers,  plotted 
on  a  large  map.  The  Government  crop  reports  de- 
scribe the  variations  of  weather  during  the  interval 
covered  in  their  crop  survey.  The  leading  newspapers 
give  daily  reports  of  the  ''Weather  in  the  Cotton  Belt." 
To-day,  August  18,  1916,  the  New  York  Times  prints 
a  typical  description  of  the  influence  of  weather  re- 
ports on  trading  and  prices : 

COTTON  ADVANCES   IN   STEADY  MARKET 

STORM  IN  THE  GULF  OF  MEXICO  KEEPS  THE  TALENT 
GUESSING 

BUT   RAINS   HELP  TEXAS   CROP 

"There  was  a  steady  undertone  in  the  cotton  market  yesterday, 
but  trading  was  rather  light  when  the  wide-spread  interest  in  cotton 
at  this  season  is  taken  into  consideration.  There  was  a  manifest 
disposition  to  wait  for  further  crop  developments,  and  the  fact  that 
there  was  a  storm  in  the  Gulf  of  Mexico  working  toward  the  cotton 
belt  made  both  the  longs  and  the  shorts  a  bit  timid.  On  one  hand 
there  was  the  possibility  that  this  disturbance  might  bring  much 
needed  rain  to  the  region  west  of  the  Mississippi;  on  the  other  hand 
the  danger  that  it  might  give  bad  weather  to  the  Southern  Atlantic 
States,  where  there  has  been  severe  damage  by  stomis  and  too  much 
rain.  .  .  . 

"Private  reports  from  the  belt  were  not  particularly  bullish,  as 
they  told  of  scattered  showers  in  Texas  and  in^pro^'ement  in  some 
parts  of  the  Eastern  States.  The  talk  of  the  approaching  storm, 
however,  i-ather  overshadowed  the  reports  of  the  weather  of  the 


Introduction  7 

minute,  although  the  bulls  did  not  neglect  to  call  attention  to  the 
fact  that  there  was  no  relief  in  Oklahoma,  where  rain  was  much 
needed." 

A  series  of  critical  questions  is  suggested  by  the  great 
importance  which  Government  Bureaus,  Cotton  Ex- 
changes, and  the  PubUc  Press  very  obviously  attach 
to  the  weather  conditions  as  they  are  related  to  the 
cotton  crop : 

(1)  Variations  of  both  temperature  and  rainfall  must 

affect  the  yield  per  acre  of  cotton,  but  do  they 
affect  the  yield  in  the  same  way  and  to  the 
same  degree?  What  is  the  measure  of  the  ef- 
fect of  each,  independent  of  the  other?  What 
is  the  measure  of  their  joint  effect? 

(2)  In  Texas,  throughout  a  quarter  of  a  century,  the 

yield  of  cotton  has  been  steadily  falhng,  while 
in  Georgia,  throughout  the  same  interval, 
the  yield  has  been  steadily  increasing.  How, 
then,  can  one  measure  the  effects,  jointly  and 
separately,  of  rainfall  and  temperature  upon 
the  crop  of  each  state  and  upon  their  combined 
crop? 

(3)  Suppose  that  the  above  questions  are  satisfac- 

torily answered  for  one  particular  month. 
Would  the  answers  be  different  for  different 
months?  Or  would  the  particular  combina- 
tion of  temperature  and  rainfall  for,  say, 
July,  produce  an  equal  effect  with  the  same 
combination  for  August?  Are  the  answers 
for  this  and  the  above  questions  the  same  for 
all  of  the  cotton-producing  states? 

(4)  Supposing  that  one  has  solved  the  above  ques- 


8         Forecasting  the  Yield  and  the  Price  of  Cotton 

tions  for  all  of  the  states  of  the  Cotton  Belt, 
how  could  one  take  account  of  the  variations 
of  the  weather  from  the  beginning  of  the  grow- 
ing season  up  to  a  given  date,  in  such  a  way 
as  to  be  able  to  forecast  their  possible  joint 
effects  upon  the  ultimate  yield  of  cotton? 
(5)  Supposing  that  one  could  forecast  the  yield  per 
acre  of  cotton  from  the  successive  reports  of 
the  Weather  Bureau,  how  would  the  degree 
of  accuracy  of  such  forecasts  compare  with 
the  forecasts  of  the  Crop-Reporting  Board, 
which  are  based  upon  the  direct  observations 
of  the  thousands  of  correspondents  of  the 
Department  of  Agriculture? 

A  knowledge  of  the  acreage  and  of  the  probable 
yield  per  acre  of  cotton  will  afford  the  necessary  data 
to  compute  the  probable  supply.  But  in  order  to  fore- 
cast the  probable  price  of  cotton  lint,  the  law  of  demand 
for  cotton  must  be  known.  That  is  to  say,  one  must 
know  the  probable  variation  in  the  price  that  will 
accompany  a  computed  variation  in  the  supply. 

With  regard  to  this  question  of  demand  economic 
science  is  in  the  state  which  electrical  science  had 
reached  about  the  middle  of  the  nineteenth  century. 
It  would  appear  that  there  are  two  sciences  of  eco- 
nomics, one  of  the  class  room  and  one  of  the  market 
place,  and  the  difference  between  the  two  is  the  same 
as  the  difference  described  by  Fleeming  Jenkin  as 
existing  between  the  Electricity  of  the  Schools  and  the 
Electricity  of  the  Practical  Engineer : 

"The  difference  between  the  Electricity  of  Schools 


Introduction  9 

and  of  the  testing  office  has  been  mainly  brought  about 
by  the  absolute  necessity  in  practice  for  definite  meas- 
urement. The  lecturer  is  content  to  say,  under  such 
and  such  circumstances,  a  current  flows  or  a  resistance 
is  increased.  The  practical  electrician  must  know  how 
much  current  and  how  much  resistance,  or  he  knows 
nothing." 

The  Open  Sesame  to  academic  economics  is  the  ''law 
of  supply  and  demand"  or  "the  equation  of  demand 
and  supply."  No  general  problem  within  the  confines 
of  the  science  may  be  approached  except  through  the 
"\sLW  of  supply  and  demand."  -But,  as  incredible  as 
it  may  seem,  what  the  law  of  demand  actually  is  for 
any  one  commodity  is  nowhere  stated  in  the  text-books. 
Indeed  not  only  do  the  text-book  writers  forbear  to 
state  the  law  for  any  one  commodity,  but,  as  a  rule, 
they  either  omit  to  say  whether  there  is  any  hope  of 
ever  knowing  the  law  in  any  concrete  case,  or  else  say 
bluntly  that  the  law  can  never  be  known  because  their 
discussion  of  economic  theory  is  confined  to  normali- 
ties within  an  hypothetical,  static  state.  The  economist 
of  the  market  place,  however,  not  only  must  know  that, 
under  given  circumstances  of  the  supply,  the  price 
will  rise  or  fall,  but  he  must  know  the  probable  limits 
within  which  the  price  fluctuations  will  be  confined. 

In  different  ways  vaSLny  agencies,  public  and  private, 
assemble  facts  that  have  a  bearing  upon  the  probable 
demand  for  cotton,  and  the  findings  of  the  several 
inquiries  are  published  for  the  information  of  those 
directly  or  indirectly  concerned.  Each  individual  is 
left  free  to  draw  his  own  conclusions  as  to  the  joint 
effect  of  the  many  factors  in  the  problem,  and  the  re- 


10       Forecasting  the  Yield  and  the  Price  of  Cotton 

suiting  conduct  of  the  many  buyers  gives  definiteness 
to  the  law  of  demand  for  cotton.  Would  it  not  be 
possible  to  describe  this  resulting  law  of  demand  with  a 
degree  of  precision  as  great  as  the  accuracy  with  which, 
from  elaborate  Government  reports  as  to  crop-condi- 
tions and  crop-prospects,  the  official  Bureau  forecasts 
the  probable  supply  of  cotton? 

By  means  of  the  principles  and  methods  presently 
to  be  described,  it  is  possible  for  any  person  (1)  from  the 
current  reports  of  the  Weather  Bureau  as  to  rainfall  and 
temperature  in  the  states  of  the  Cotton  Belt,  to  fore- 
cast the  yield  of  cotton  with  a  greater  degree  of  accuracy 
than  the  forecasts  of  the  Department  of  Agriculture, 
and  (2)  from  the  prospective  magnitude  of  the  crop, 
to  forecast  the  probable  price  per  pound  of  cotton  with 
a  greater  precision  than  the  Department  of  Agriculture 
forecasts  the  yield  of  the  crop. 

The  principal  purpose  of  my  Essay  I  should  Uke  to 
make  very  clear.  It  is  not  to  point  out  the  limitations 
of  the  work  done  in  forecasting  by  the  Department  of 
Agriculture;  much  less,  to  urge  any  device  of  my  own 
as  a  substitute  for  the  methods  that  are  followed  by  the 
official  Statistical  Bureaus.  My  chief  aim  has  been 
to  make  a  contribution  to  economic  science  by  showing 
that  the  changes  in  the  great  basic  industry  of  the  South 
which  dominate  the  whole  economic  life  of  the  Cotton 
Belt  are  so  much  a  matter  of  routine  that,  with  a  high 
degree  of  accuracy,  they  admit  of  being  predicted  from 
natural  causes. 

The  business  of  economic  science,  as  distinguished 


Introduction  11 

from  economic  practice,  is  to  discover  the  routine  in 
economic  affairs.  It  aims  to  separate  out  the  elements 
of  the  routine,  to  ascertain  their  interdependence,  and 
to  use  the  knowledge  of  their  connections  to  anticipate 
experience  by  forecasting  from  known  changes  the 
probabilities  of  correlated  changes.  The  seal  of  the 
true  science  is  the  confirmation  of  the  forecasts;  its 
value  is  measured  by  the  control  it  enables  us  to  exer- 
cise over  ourselves  and  our  environment. 


CHAPTER   II 

THE  MATHEMATICS  OF  CORRELATION 

"The  true  Logic  for  this  world  is  the  Calcuhis  of  Probabihties,  the 
only  Mathematics  for  Practical  Men." 

—  James  Clerk  Maxwell. 

In  an  Announcement  ^  issued  April  29,  1916,  by  the 
Office  of  Markets  and  Rural  Organization  of  the  U.  S. 
Department  of  Agriculture,  there  is  a  ^'Review  of  Some 
of  the  Provisions  of  the  Pending  Cotton  Futures  Bill, 
H.  R.  11861,  and  of  Causes  of  Differences  Between  Prices 
of  Middling  Cotton  in  New  York  and  Liverpool. "  Three 
valuable  charts  are  given  of  fluctuations  in  different 
markets  of  prices  of  spot  cotton  and  prices  of  cotton 
futures.  One  of  the  charts  is  described  in  these  words: 
"Chart  3  shows  the  variation  in  the  prices  of  futures  -  on 
the  cotton  exchanges  at  New  York  and  New  Orleans, 
as  compared  with  the  price  of  Middling  as  determined 
by  averaging  the  quotations  obtained  from  the  desig- 
nated spot  markets,  as  follows:  Norfolk,  Augusta, 
Savannah,  Montgomery,  New  Orleans,  Memphis,  Little 
Rock,   Dallas,    Houston,    and   Galveston.     The   chart 

^  Service  and  Regulatory  Announcements,  No.  9. 

2  The  meaning  of  "futures"  throughout  the  investigation  is  given  in 
a  description  of  the  statistical  data:  "The  future  quotation  for  each  day 
is  always  that  for  contracts  which  are  to  be  fulfilled  in  the  current  month. 
During  the  last  five  days  of  a  month,  when  contracts  for  the  present 
month  are  no  longer  traded  in,  contracts  for  the  following  month  are 
substituted,  as  they  may  be  considered  essentially  the  current  month, 
for  such  contracts  may  be  purchased  or  sold  and  immediately  fulfilled 
or  closed."     Ibid.,  p.  101. 


The  Mathematics  of  Correlation  13 

covers  the  time  between  February  15,  1915,  and  Jan- 
uary 22,  1916."  From  a  study  of  this  and  the  other 
two  charts,  the  writer  of  the  Review  concludes  ''that 
since  the  cotton  futures  Act  ^  went  into  operation  future 
quotations  have  fairly  reflected  spot  values  in  both 
New  York  and  New  Orleans,  and  also  in  a  general  way 
over  the  entire  South,  and  that  the  law  has  thus  ac- 
complished and  is  accomplishing  the  end  for  which  it 
was  enacted."    Ihid.^  p.  104. 

With  the  question  as  to  whether  the  cotton  futures 
Act  is  doing  the  work  for  which  it  was  enacted,  we  are 
not  at  present  concerned,  but  we  are  interested  in  the 
statement  of  fact  that  "since  the  cotton  futures  Act 
went  into  operation  future  quotations  have  fairly  re- 
flected spot  values  in  both  New  York  and  New  Orleans, 
and  also  in  a  general  way  over  the  entire  South."  The 
official  words  are  that  ''future  quotations  have  fairly  re- 
flected spot  values. ' '  Just  what  is  meant  hy  fairly?  How 
can  one  measure  the  degree  of  association  between  futures 
and  spot  values?  Or,  to  put  the  question  in  another 
form,  suppose  one  knew  the  probable  spot  values  in  the 
South,  how  could  one  forecast  the  price  of  futures  on 
the  cotton  exchanges  at  New  York  and  New  Orleans? 
These  are  types  of  problems  which  the  statistical  meth- 
ods we  are  about  to  describe  enable  us  to  solve.  Let 
us  propose  a  definite  problem  and  connect  the  exposi- 
tion of  the  statistical  methods  with  the  solution  of  the 
problem : 

On  Figure  1,  the  two  graphs  record  for  an  interval 
of  42  days,  from  September  11  to  October  30,  1915, 
the  fluctuating  prices  of  average  spots  in  the  South  on 

'  The  Act  of  1914. 


14      Forecasting  the  Yield  and  the  Price  of  Cotton 


-\ 1 1 r 


"1       I       I       1       I       1       r 


o 

>H 

> 

<u 

b 

n 

^i^ 

r/i 

Cf 

"1 

B 

\ 

d 

% 

o 

<1 

I 

•0 


^ 
? 


UO^OJ y.O    S-30U(J 


The  Mathematics  of  Correlation  15 

the  ten  markets  which  were  enumerated  above,  and  the 
fluctuating  prices  of  futures  on  the  New  York  exchange. 
The  general  trend  of  each  series  of  figures  shows  an 
ascent  to  about  the  middle  of  the  record  and  then  a 
descent.  Suppose  we  were  to  make  allowance  for  the 
general  trend  in  the  two  graphs,  what  would  be  the 
degree  of  connection  between  the  fluctuations  of  the 
futures  from  their  general  trend  and  the  fluctuations 
of  the  spots  from  their  general  trend?  To  be  more 
definite,  suppose  we  represent  the  general  trend  of 
each  series  of  figures  by  a  progressive  a\'erage  of  five 
daily  quotations;  that  is  to  say,  suppose  we  place  on 
both  series  for  each  day  a  mark  indicating  the  mean 
of  the  respective  quotations  for  the  five  days  of  which 
the  given  day  is  the  middle  day.  We  should  then  ob- 
tain for  each  series  a  number  of  points  that  would 
indicate  its  general  trend.  If  we  take  the  fluctuations 
of  each  series  from  its  own  general  trend,  we  shall  have 
the  data  for  the  problem  which  we  propose  to  solve, 
namely,  to  ascertain  the  degree  of  association  between 
the  fluctuations  of  futures  and  the  fluctuations  of 
spots. 

Figure  2  shows  the  actual  quotations  for  futures 
on  the  New  York  exchange  and  the  general  trend 
of  the  figures  when  the  general  trend  is  derived 
from  a  progressive  average  of  five  daily  quota- 
tions. Figures  1  and  2  exhibit  data  for  only  42 
days.^ 

^  The  data  used  by  the  Government  office,  covering  records  for  275 
to  280  days,  were  kindly  suppHed  to  me  by  Mr.  Charles  J.  Brand,  Chief 
of  the  Office  of  Markets  and  Rural  Organization.  When  we  come  to  the 
apphcation  of  our  statistical  methods  we  shall  use  all  of  the  available 
data. 


16       Forecasting  the  Yield  and  the  Price  of  Cotton 


if-io^  Ma/\/  ui  s-9jn^nj.  uo^oo^o  Q^yUf-j 


The  Mathematics  of  Correlation  17 

We  pass  now  to  the  development  of  the  mathematical 
theory  of  correlation.^ 

A  Frequency  Distribution 

Statistical  tables  that  show  either  the  absolute  or 
relative  frequencies  of  observations  for  given  types  of 
measurements  are  called  frequency  tables,  or  frequency 
distributions.  The  accompanying  Table  1  is  a  fre- 
quency distribution  showing  the  absolute  frequencies  in 
the  fluctuations  of  the  average  prices  of  spots  from  their 
general  trend,  the  general  trend  being  deri\'ed  from  a 
progressive  five  days  average. 

After  the  raw  observations,  for  purposes  of  facility 
in  the  handling  of  the  data,  ha\'e  been  grouped  into 
appropriate  frequency  distributions,  the  next  step  is 
to  describe  the  distributions  by  the  aid  of  the  fewest 
possible  measurements  that  will  enable  one  to  summar- 
ize the  features  of  thq.  distribution  which,  for  the  pur- 
pose in  hand,  are  most  important. 

One  of  the  most  important  summary  descriptions 
of  a  frequency  distribution  is  the  mean  value  of  the 
distribution.  In  the  particular  problem  before  us  the 
mean  A-alue  of  the  fluctuations  of  average  spots  from 
their  general  trend  is  the  quantity  that  we  wish  to 
ascertain.  This  brings  us  to  the  first  step  in  our  math- 
ematical work. 

^  I  wish  most  gratefully  to  thank  Professor  Karl  Pearson  for  the  in- 
struction that  I  received  in  his  laboratory  several  years  ago,  anil  for 
the  inspiration  of  his  published  works.  To  him,  almost  exclu.«ively, 
I  owe  my  knowledge  of  the  theor}'  of  correlation.  In  b(>ginning  the 
study  of  Professor  Pearson's  writings,  I  received  help  fioin  Professor 
G.  U.  Yule's  article  "On  the  Theory  of  (>)rrelation,"  in  th"  Jotinial 
of  the  Royal  Statistical  Society,  December,  bS97.  and  rrnm  Mf.  \\  . 
Palin  Elderton's  treatise  on  Frequency  Curves  mrl  Corn'l.ilr)n. 


18       Forecasting  the  Yield  and  the  Price  of  Cotton 

TABLE  1.  —  Frequency  Distribution  of  Fluctuations  op  the 
Prices  of  Average  Spots  from  a  Five  Days  Progressive 
Mean  of  Prices 


Fluctuations  of  Average 
Spots  (Cents) 

Frequency 

(Number  of  Days 

on  which  the 

Fluctuations 

Occurred) 

—  .165 

to 

—  .135 

3 

—  .135 

to 

—  .105 

3 

—  .105 

to 

—  .075 

4 

—  .075 

to 

—  .045 

23 

—  .045 

to 

—  .015 

55 

—  .015 

to 

+  .015 

107 

+  .015 

to 

+  .045 

54 

+  .045 

to 

+  .075 

16 

+  .075 

to 

+  .105 

7 

+ .  105 

to 

+  .135 

2 

+  .135 

to 

+  .165 

1 

Total 

275 

The  Mathematics  of  Correlation 


19 


Theorem  I.  The  algebraic  sum  of  the  deviations  of  a 
series  of  magnitudes  from  their  arithmetical  mean  value 
is  zero. 

Let  the  magnitudes  be  Xi,  Xt,  Xz,  .  .  .  Xn,  N  in  number, 
and  let  their  arithmetical  mean  value  be  x.  Then,  by 
the  definition  of  the  arithmetical  mean,  we  have 

.1*1  +  Xo  +  X3  +  .  .  .  Xfi 


N 
N  X  =  Xi  +  Xo  +  X3  + 


^nf 


and  {xi  —  x)  -\-  (.To  —  x)  +  (xs  ~  x)  -\-  .  .  .  {x„  —  x)  =0. 
But  the  quantities  on  the  left-hand  side  of  the  equation 
are  the  deviations  of  the  magnitudes  from  the  arith- 
metical mean  of  the  magnitudes,  and  the  sum  of  these 
deviations  is  proved  to  be  zero.  This  theorem  we  shall 
use  later  on  in  our  work. 

Theorem  II.  The  arithmetical  mean  of  a  series  of  mag- 
nitudes is  equal  to  any  arbitrary  quantity  plus  the  mean 
of  the  deviations  of  the  magnitudes  from  the  arbitrary 
quantity. 

As  before,  let  the  magnitudes  be  xi,  x^,  Xz,  .  .  .  x„,  and 
let  P  be  the  arbitrary  quantity. 


Then  x 

the  x's. 


Also  we  have 


where  S(.t)   is  put  for  the  sum  of 


Xi  =  P  +  x[,  where  x[  is  the  deviation  of  Xi  from  P; 


X2  =  P  -{-  Xo,  where  x^ 
Xz  =  P  +  x'^,  where  x'^ 


x„  =  P  +  x',„  where  x,' 


X2  from  P; 
Xz  from  P; 


x„  from  P. 


20       Forecasting  the  Yield  and  the  Price  of  Cotton 

Therefore  S(x)  =  A^  P  +  S(xO,  and  ?^L  p  +  ?^ 

iV  A 

which  is  the  proposition  we  had  to  prove. 

We  shall  now  apply  this  latter  theorem  to  find  the 
mean  value  of  the  fluctuations  of  average  spots  from 
their  general  trend.    The  data  are  given  in  Table  2. 

Here,  the  arbitrary  quantity  from  which  the  fluctua- 
tions are  measured  is  zero.  Column  II  gives  the  fluc- 
tuations measured  from  zero  expressed  in  terms  of  the 
unit  of  grouping.  According  to  Theorem  II,  the  arith- 
metical mean  is  equal  to  the  arbitrary  quantity  plus  the 
mean  of  the  deviations  from  the  arbitrary  quantity. 
Consequently,  in  this  particular  case,  the  arithmetical 
mean  of  the  price  fluctuations  is  (—.07)  in. units  of 
grouping,  or  (—  .002)  in  absolute  units. 


The  Standard  Deviation  as  a  Measure  of  Dispersion 

The  arithmetical  mean  of  the  frequency  distribution 
gives  us  one  of  the  most  important  summary  descrip- 
tions of  the  distribution:  it  gives  the  centre  of  density 
of  the  distribution.  But  in  economic,  as  well  as  in  most 
other,  measurements  it  is  extremely  important  to  know 
how  the  several  observations  are  grouped  about  the 
arithmetical  mean  of  the  measurements,  and  a  co- 
efficient showing  the  manner  of  grouping  is  a  measure 
of  dispersion.  Just  as  we  found  that  the  arithmetical 
mean  of  the  measurements  gives  us  an  idea  of  the 
centre  of  the  density  of  the  measurements,  so,  as  a 
measure  of  dispersion,  we  might  take  the  arithmetical 
mean  of  the  deviations  of  the  magnitudes  from  the 
mean  of  the  observations.     But  if  we  followed  this 


The  Mathematics  of  Correlation 


21 


TABLE  2.  • — •  Computation  of  the  Mean  of  the  Fluctuations  of 
Average  Spots  from  Their  General  Trend 


I 

Fluctuations 

of 

Average   Spots 

II 

Fluctuations 

Expressed  in  Units 

of  Grouping 

Unit  =  .  03 

x' 

III 

Frequency 
/ 

IV 

Product  of 

Column  II  by 

Column  III 

fx' 

—  .15 

—5 

3 

—  15 

—  .12 

—4 

3 

—  12 

—  .09 

—3 

4 

—  12 

—  .06 

—2 

23 

—  46 

—  .03 

—1 

55 

—  55 

0 

107 

+  .03 

+  1 

54 

+  54 

+  .06 

+2 

16 

+  32 

+  .09 

+3 

7 

+  21 

+  .12 

+4 

2 

+     8 

+  .15 

+5 

1 

+     5 

Totals 

275 

—140 
+  120 

—  20 

The  mean  fluctuation  from  the  general  trend  is,  therefore, 


—20 


275 


=  —.07 


in  units  of  grouping,  or  ( — -.07)  (.03)  =  — .002  in  absolute  units. 


plan,  we  should  meet  with  an  embarrassing  difficulty: 
The  deviations  of  the  measurements  from  the  arith- 
metical mean  are  some  of  them  positive  and  some  of 
them  negative,  and  if  we  take  account  of  the  signs  of 


22       Forecasting  the  Yield  and  the  Price  of  Cotton 

the  deviations,  then,  according  to  Theorem  I,  the  sum 
of  the  deviations  is  zero.  We  therefore  choose,  as  our 
measure  of  dispersion,  the  square-root  of  the  mean 
square  of  the  deviations  about  the  arithmetical  mean 
of  the  observations,  and  we  call  this  measure  of  dis- 
persion the  standard  deviation. 

If  we  let  (J  represent  the  standard  deviation,  then,  if 

X2  —  X  =  A2, 

Xz      X  =  A  3, 

we  shall  have  as  the  symbolic  expression  of  the  stand- 
ard deviation 


^1 


N 


Theorem  III.  The  square  of  the  standard  deviation 
of  a  series  of  magnitudes  is  equal  to  the  mean  square 
of  the  deviations  of  the  magnitudes  about  an  arbitrary 
quantity,  minus  the  square  of  the  difference  between  the 
arbitrary  quantity  and  the  mean  of  the  magnitudes. 

As  before,  let  the  quantities  be  Xi,  Xi,  Xz,  .  .  .  Xn  and 
their  mean  value  be  x.  Let  the  arbitrary  quantity 
be  P  and  let  the  difference  between  the  arbitrary  quan- 
tity and  the  mean  be  dx,  so  that  x  =  P  -{-  dx.  Let  the 
deviations  of  the  quantities  from  the  arithmetical  mean 
be  Xi,  X2,  X3,  -  •  '  Xn  and  their  deviations  from  P  be 
x[,  X2,  X3,  .  .  .  x'„.    We  shall  then  have 


The  Mathematics  of  Correlation  23 


Jl    1       =        XX      Xy 

Ji.2   ^=   X2  X, 

(1)  <    Xz  =   Xs-X, 

(2)  P  =  x~  4; 


(3) 


(4) 


'  x[  =  Xi  —  P  =  Xi~ (x  —  d^)  =  (xi—x)  +d^  =  Xi  +d^, 
X2  ^^  X2 — -I  =^  X2  —  (X  —  a^j  ^  (S^o  —  •^/  ~l~ ^x  ^^  -^ 2  ~t~ '^xj 
Xs  =  Xs  —  P  ^  x^—  (x  —  d^)  =  (x^—x)  +d^  =  X^+d^, 

^n  —  ^n        P—^n         (^         d^)  =  {X^       X)  +  d^  =  X„  +  Ctj;/ 

'  (:r;)2  =  (X,  +  d^y  =  X^  +  2  4  Xi  +  cTi, 

(a:;)2  =  (X3  +  d^y  =  XI  +  2  4  X3  +  dj, 
I  (x:)'^  =  (X„  +  4)^  =  X?.  +  2  4  X„  +  d^i; 


(5)  Therefore  S(x')-  =  ^(X^)  +  2(^^(X)  +  Nd^. 

But    according   to    Theorem   I,    S(X)  =  zero,    and, 

S(X^)       SM^        .       ^.  ,    .     , 

consequently  — tt —  =   ~Tj~  —  «i.      Since    (7i   is   by 

definition  equal  to   — ^tj—,    we   have   cri  =  — a7^~«x> 

which  was  to  be  proved. 

Corollary.  The  mean  square  deviation  about  the  arith- 
metical mean  of  the  observations  is  less  than  the  mean 
square  deviation  about  any  arbitrary  quantity. 

TTr  ,       .  1  ,     2(X2)    i:(x'y    ^ 

We  have  just  proved  that      ^     =  — i^  —  dx. 

The  left-hand  side  of  the  equation  is  a  positive  quantity 
because  it  is  a  mean  square.    The  right-hand  side  must, 


24       Forecasting  the  Yield  and  the  Price  of  Cotton 

therefore,  also  be  a  positive  quantity,  but  it  consists 
of  the  difference  between  two  positive  quantities,  the 
greater  of  which  is  the  mean  square  deviation  about  an 
arbitrary  quantity.  The  same  equation  would  hold  no 
matter  what  the  arbitrary  quantity  might  be.  There- 
fore the  mean  square  deviation  about  the  arithmetical 
mean  is  less  than  the  mean  square  deviation  about  any 
arbitrary  quantity. 

We  shall  now  use  this  theorem  to  calculate  the  value 
of  (Tx  for  the  fluctuations  of  the  average  prices  of  spot 
cotton  from  the  general  trend  of  prices.  The  data  are 
given  in  Table  3. 

Our  mathematical  theory  of  correlation  is  developed 
as  an  instrument  to  forecast  economic  events.  We  may 
stop  for  a  moment,  therefore,  to  consider  the  bearing 
of  our  results  thus  far  upon  the  problem  of  forecasting. 

Figure  3  shows  a  smooth  curve  ^  passing  closely  to  the 
broken  hne  representing  the  frequency  distribution  of 
the  fluctuations  of  average  spot  prices  from  their  gen- 
eral trend.  This  curve  is  a  synmietrical  curve  in  the 
sense  that  the  two  sides  of  the  figure  are  similarly  dis- 
posed with  reference  to  the  maximum  ordinate.  If, 
for  instance,  the  right-hand  side  of  the  figure  were 
made  to  revolve  about  the  maximum  ordinate  and  be 
placed  upon  the  left-hand  side,  the  two  parts  of  the 
curve  would  be  congruent.  This  symmetrical  curve 
is  called  the  normal,  or,  sometimes,  the  Gaussian  curve, 
after  the  author  of  Theoria  Motus  Corporum  Coelestium, 
who  was  one  of  the  first  to  investigate  its  properties. 
If  we  represent  by  x  the  deviations  of  the  abscissas  from 

1  In  fitting  the  smooth  curve  to  the  data,  the  value  of  <r  was  computed 
with  Sheppard's  correction. 


The  Mathematics  of  Correlation 


25 


TABLE  3.  —  Computation   of  the   Standard   Deviation   of  the 
Fluctuations  of  Average  Spots  from  the  General  Trend 


I 

Fluctuations 

of 
Average  Spots 

II 

Fluctuations 

Expressed  in 

Units  of 

Grouping 

Unit  = .  03 

x' 

III 

Frequency 
/ 

IV 

Product  of 

Column  II  by 

Column  III 

fx' 

V 

(x'P 

VI 

Kxr 

—  .15 

—  5 

3 

—  15 

25 

75 

—  .12 

—  4 

3 

—  12 

16 

48 

—  .09 

o 
—  O 

4 

—  12 

9 

36 

—  .06 

—  2 

23 

—  46 

4 

92 

—  .03 

—  1 

55 

—  55 

1 

55 

0 

107 

+  .03 

+  1 

54 

+  54 

1 

54 

+  .06 

+  2 

16 

+  32 

4 

64 

+  .09 

+  3 

7 

+  21 

9 

63 

+  .12 

+  4 

2 

+    8- 

16 

32 

+  .15 

+  5 

1 

+    5 

25 

25 

Totals 

275 

—  140 
+  120 

544 

—  20 

According  to   the  sjonbols  in  the  text  dx  is  the  mean   deviation 
from  the  arbitrary  origin,  and,  consequently,  in  this  particular  case, 

''^  ~  275  '' 


.0727,  and  dl  =  .005285.     The  mean  square  deviation 


about  the  arbitrary  origin  is 

^^^!    "x 


N 


KAA 

1^  =  1.978182.    By  Theorem 


ju-      —       lu-  d^,  and,  consequently,  cr^  =  1.978182  — 

.005285  =  1.972897.    Therefore  cr^  =  \/l- 972897  =  1.405  in  units  of 
grouping,!  or  (1.405)  (.03)  =  .042  in  absolute  units. 


1  The  value  of  o-j.  is  here  derived  from  the  raw  data,  but  in  frcquencj' 
curves  of  high  contact,  that  is  to  say,  in  curves  which  tail  off  in  the 

manner  of  Figure  3,  the  value  of  o-^  is  —rj r^-     Tliis  correction, 

which  is  known  as  Sheppard's  correction,  I  have  deliberately  omitted 
because  I  wish  to  present  the  theory  of  correlation  only  in  its  bold 
outlines. 


26        Forecasting  the  Yield  and  the  Price  of  Cotton 


no 

— 1 —  I 

,,..,     . 

1 

1              1  - 

- 

lOO 

- 

/ 

\ 

- 

90 

- 

/ 

\ 

- 

so 

- 

I 

4 

- 

70 

- 

I 

\ 

- 

1" 

- 

1 

\ 

- 

*^    JO 

\\ 

-«7 

- 

\\ 

- 

30 

- 

i 

'/ 

\\ 

- 

20 

- 

/ 

\ 

- 

lO 

- 

^ 

/ 

V. 

- 

L_ 

.;^ 

(      1          r 

r: 

,-T 

1         1     1 

X^^ 

3 

-./S        -.12        -.09        -.06         -03  O         +.OJ         +06        +.09        +.IB         +.15 

Fluctuations  ofaven"^  jpofj  -from  pro^ss/i'e  Sdoyj  means. 

Figure  3.  —  The  frequency  distribution  of  the  fluctuations  of  average 
spot  prices  from  their  general  trend. 

Equation  to  the  smooth  curve,  y  =  79.81e      .0034. 


The  Mathematics  of  Correlation  27 

the  mean  value  of  the  distribution,  and  by  y  the  ordinate 

corresponding  to  x,  then  the  equation  to  the  normal 

N      _£i 
curve  IS  y  =  — 7=  e    2<t^-,  where  A^  is  the  number  of 
cr\/2ir 

the  observations,  a  is  the  standard  deviation  of  the 

observations,   tt  is  the  ratio  of  the  circumference  of 

a  circle  to  its  diameter  and  is  equal  to  3.1416,  and 

e  is  the  base  of  Naperian   logarithms  and  has  the 

value  2.7183.    If  the  distribution  of  the  data  is  normal, 

all  that  is  required  to  get  the  equation  to  the  smooth 

Gaussian  curve  fitting  the  data  is  to  substitute  for  N 

and  0-,  in  the  above  equation,  the  values  obtained  for 

these  constants  from  the  concrete  data  of  the  problem. 

From  an  investigation  of  the  properties  of  the  normal 
curve,  it  is  found  that  in  a  perfectly  normal  distribu- 
tion of  data,  68  per  cent  of  all  the  observations  fall 
within  +  a;  that  is  to  say,  68  per  cent  of  the  observa- 
tions deviate  less  than  plus  a  or  minus  cr  from  the 
mean  value  of  the  frequency  distribution.  Further- 
more, 95  per  cent  of  all  the  observations  fall  within 
±  2o-,  and  99.7  per  cent  of  all  the  observations  fall 
within  +  3o-.  On  Figure  3,  the  distances  of  o",  2<j,  Za 
from  the  mean  value  of  the  distribution  are  indicated 
by  special  marks.  ^ 

We  may  now  see  why  it  was  desirable  to  describe  a 

1  As  the  normal  curve  is  used  so  constantly  in  matheniatical  compu- 
tations, tables  have  been  constructed  showing  the  proportion  of  cases 
included  between  the  mean  and  a  deviation  from  the  mean  of  any 
multiple  or  submultiple  of  the  standard  deviation.  One  of  the  best 
compilations  is  Sheppard's  Tahlen  of  the  ProbabilUy  Integral  which  is 
contained  in  Professor  Pearson's  Tables  for  Statisticians  and  Biometri- 
cians.  By  the  aid  of  this  Table,  after  we  have  computed  the  mean 
and  standard  deviation  of  a  given  normal  distribution,  we  can  obtain 
the  probability  of  any  deviation  from  the  mean  value. 


28       Forecasting  the  Yield  and  the  Price  of  Cotton 

frequency  distribution  by  its  mean  and  its  standard 
deviation.  In  any  system  of  forecasting  economic 
events,  it  is  clearly  of  first  importance  to  predict  the 
events  with  the  greatest  possible  precision,  which  is 
equivalent  to  reducing  as  far  as  possible  the  scatter  or 
standard  deviation  of  the  predicted  events.  By  taking 
the  arithmetical  mean  about  which  to  measure  the 
scatter,  we  have  in  the  standard  deviation,  according 
to  the  Corollary  of  Theorem  III,  a  measure  of  disper- 
sion which  is  less  than  the  root-mean-square  deviation 
about  any  other  arbitrary  quantity.  Moreover,  by  the 
help  of  the  Tables  of  the  Probability  Integral,  when  we 
use  a  as  the  measure  of  scatter,  we  have  at  once,  in 
case  our  frequency  distributions  are  approximately 
normal,  a  numerical  measure  of  the  precision  of  our 
forecasts.  This  point  will  be  illustrated  further  on  in 
our  work. 

The  Fitting  of  Straight  Lines  to  Data 

We  have  thus  far  described  the  mean  and  the  stand- 
ard deviation  of  a  simple  frequency  distribution,  and 
the  particular  distribution  which  we  used  as  an  illus- 
tration was  the  fluctuation  of  the  average  prices  of  spot 
cotton  in  the  South  from  the  general  trend  of  spot 
prices,  the  general  trend  being  derived  from  a  five  days 
progressive  mean.  What  we  did  for  the  spot  prices  we 
could  do  for  the  prices  of  futures  on  the  New  York 
exchange.  We  should  then  be  brought  a  step  nearer 
to  our  concrete  problem,  which  is  to  measure  the  de- 
gree of  correlation  between  the  fluctuations  of  New  York 
futures  and  the  fluctuations  of  average  spots  in  the 
South. 


The  Mathematics  of  Correlation  29 

Figure  4  displays  the  relation  between  the  fluctuations 
of  New  York  futures  and  the  fluctuations  of  average 
spots  in  the  South.  A  diagram  like  Figure  4  showing 
the  relation  between  the  variations  of  two  variables 
is  called  a  scatter  diagram.  From  the  general  sweep 
of  the  scatter  diagram,  it  is  clear  that,  as  the  fluctuations 
of  average  spots  increase  or  decrease,  the  fluctuations 
of  New  York  futures  increase  or  decrease.  There  is 
obvious  association  between  the  two  variables,  and  the 
substance  of  our  problem  is  to  measure  the  degree  of 
the  correlation  and  to  find  the  statistical  law  descrip- 
tive of  the  manner  in  which  the  one  variable  is  connected 
with  the  other. 

On  the  scatter  diagram  of  Figure  4  the  observations 
are  represented  by  points.^  Figure  5  describes  the  data 
of  Figure  4  in  a  way  to  exhibit  more  clearly  the  nature 
of  the  correlation  of  the  two  variables.  For  each  typical 
value  of  X  in  Figure  4  there  is  a  corresponding  array 
of  i/'s,  and  Figure  5  shows,  for  each  type-value  of  x, 
the  mean  value  of  the  corresponding  array  of  t/'s.  It 
is  clear  that  if  we  could  fit  properly  a  straight  line  to 
the  points  of  Figure  5,  the  equation  to  the  straight  line 
would  give  us  the  statistical  law  connecting  the  changes 
of  the  two  variables. 

To  cover  the  steps  in  the  development  of  our  method 
let  us  first  recall  that  the  trigometric  tangent  of  an 
acute  angle  in  a  right-angle  triangle  is  equal  to  the 
ratio  of  the  side  opposite  the  acute  angle  to  the  side 

'  There  are  275  observations,  but  as  a  pjreat  many  of  them  have  the 
same  values  some  of  the  representative  points  ai'e  superposed  upon 
others.  In  making  the  computations  that  follow  in  the  text  each  point 
is  regarded  as  having  an  importance  projxjrtionate  to  the  number  of 
observations  that  it  represents. 


30        Forecasting  the  Yield  and  the  Price  of  Cotton 


r-i— 

1 

1 I 

I          1 

1 1 \ [— 

-■  r 

"r.ZA 

- 

• 

• 

+.2/ 

- 

• 

• 

+-./6 

- 

■ 

vj 

S 

• 

S  +-./5 

- 

• 

.    . 

- 

5 

. 

*) 

.    . 

^  ^-1^ 

- 

• 

•    • 

- 

% 

.         .    • 

"0 

... 

V    'rD9 

- 

. 

- 

.51 

^ 

^ 

•      • 

\..06 

- 

.   .  . 

- 

5 

• 

<^ 

...                             • 

5  1-.OJ 

- 

• 

•    •    • 

.    .             .        « 

- 

p 

^ 

•                   •      • 

.             « 

* 

. 

2     o 

- 

. 

.   . 

- 

K. 

i 

t; 

• 

•             *            • 

• 

^-.03 

- 

. 



.   . 

- 

t 

• 

• 

:g 

♦ 

• 

.... 

i-.Q5 

-  • 



• 

.«i 

^ 

o  -.09 

- 

• 

- 

V. 

• 

•     « 

< 

O 

•*:  -.12 

- 

* 

- 

Cl 

-fi 

* 

5j 

• 

• 

•      • 

• 

t^-./5 

• 

• 

-la 

• 

• 

-^1 

-X- 

J... 

1            ' 

1           1 

1 1 1 L_ 

1 

-.15       -12      -.09     -.06      -.03         O        *.03 '  ^.06      +09      \I2       t/-? 
Fluctuations  ofoveroce  spots  from  pmqrr.iS't^e  5  days  means. 

Figure  4.  —  Scatter  diagram  showing  the  relation  between  the  fluctua- 
tions of  futures  and  of  spots. 


The  Mathematics  of  Correlation 


31 


f.eA 

— 1 1 1 1 1 1 1 1 1 1 r— 

*2I 

/         /- 

5 

/    ' 

\  '" 

/ 

/ 

S    -OS 

:              /      : 

1 

/ 

V;  ^.03 

/                 - 

futures 

0 

/ 

/ 

O    -.09 

■       r7                      - 

r 

vj>     -.15 

■     // 

/ 

-IB 

/ 

-.21 

1      1       1      1      1       1       1      1      1       1      1 

-JS        -.12        -09         -.06        -OS  O         i-.OJ        1-06        *.09        +/^        -r.lS 

F/ucfuot/ons  of  overage  spots  from  pro^ress/'i/e  S  days  /neons 

Figure  5.  —  The  statistical  law  connecting  the  fluctuations  of  futures 

and  of  spots. 

Equation  to  the  straight  line,  \j  =  1.50x  +  .002,  origin  at  (0,  0). 


32       Forecasting  the  Yield  and  the  Price  of  Cotton 


adjacent  to  the  acute  angle.    For  example,  in  the  right- 

angle  triangle,  Figure  6,  tan  a  =  7^^.     Let  us  lurtner 

recall  that  the  equation  to  a  straight  line  may  be  put 
into  the  form  y  =  mx  -\-  b,  where  m  is  the  tangent  of  the 
o  angle  which  the  line  makes  with 
the  axis  of  x,  and  b  is  the  inter- 
cept on  the  axis  of  y.  For  example, 
in  Figure  7,  DE  is  the  line  whose 
equation  is  sought.  Let  B  rep- 
resent a  point  on  the  Hne  with 
coordinates  (x,  y).  Then  y  =  BC 
■^  CF  =  AC  tan  a  +  b  =-  x  tan  a  -\-  b  =  mx  -\-  b.  In 
the  straight  line  corresponding  to  this  equation, 
y  =  mx  +  b,  the  slope  of  the  line  will  vary  with  the 
sign  and  magnitude  of  m,  and  the  position  of  the  line 
with  reference  to  the  axes  of  coordinates  will  vary 
with  the  sign  and  magnitude  of  b. 

y 


Figure  6. 


Figure  7. 


The  statistical  problem  is  to  find  the  values  of  m  and 
b  from  the  concrete  data  of  the  scatter  diagram  so  that 


The  Mathematics  of  Correlation  33 

the  resulting  straight  line  will  give  the  best  fit  to  the 
data.  The  expression  "best  fit"  is  seldom  defined. 
Its  significance  varies  with  the  problem  in  hand  and  it 
generally  means  a  fit  which  is  convenient  and  which, 
for  the  problem  to  be  solved,  gives  satisfactory  results.^ 
The  principle  upon  which  the  values  of  m  and  b  are 
determined  is  so  to  choose  m  and  b  as  to  make  the  mean 
square  deviation  of  the  observations  from  the  resulting 
straight  line  a  minimum.  The  pertinency  of  this  prin- 
ciple for  our  problem  of  forecasting  is  plain,  because  we 
have  already  learned  that  when  observations  are  dis- 
tributed according  to  the  normal  law,  the  Tables  of  the 
Probability  Integral  enable  us  to  compute  the  probability 
of  a  deviation  equal  to  any  multiple  or  submultiple  of 
the  root-mean-square  deviation.  Moreover,  as  in  all 
problems  of  forecasting  it  is  desirable  to  have  the  root- 
mean-square  deviation  as  small  as  possible,  it  is  obvious 
that  a  straight  line  which  fits  given  data  so  as  to  make 
the  mean  square  deviation  of  the  points  from  the 
straight  line  a  irdnimum  is,  for  the  problem  in  hand  of 
forecasting  one  variable  from  a  knowledge  of  the  other, 
a  good  fit  to  the  data. 

In  Figure  5  let  the  abscissas  of  the  series  of  points  be 
Xi,  Xo,  Xs,  .  .  .  ,  and  let  the  corresponding  ordinates  be 
Vx,,  Vx^y  Vx,,  ■  •  •  Each  of  the  ordinates,  we  recall,  is  the 
mean  of  the  array  of  points  in  Figure  4  corresponding  to 
the  typical  value  of  x.  Suppose  that  yx,  is  determined 
from  n^^  points;  ij^,  from  n^,  points;  y^,  from  Ux,  points; 
and  so  on,  for  the  other  values  of  y^.  Then  nx,  +  w^,  + 
fix^  -\-  .  .  .  =  N,  which  is  the  total  number  of  observa- 

1  See  "The  Statistical  Complement  of  Pure  Economics,"  Quarterly 
Journal  of  Economics,  November,  1908,  pp.  18  to  23. 


34       Forecasting  the  Yield  and  the  Price  of  Cotton 

tions.  Now  the  condition  that  the  mean  square  de- 
viation of  the  points  in  Figure  5  from  the  straight  Hne 
shall  be  a  minimum  is,  symbolically,  that 


N 


shall  be  a  minimum. 


Here  y  is  the  ordinate  of  the  straight  line;  y^  is  the 
ordinate  of  a  point  in  Figure  5  corresponding  to  ab- 
scissa x;  n^  is  the  number  of  observations  or  points  in 
the  given  array;  N  represents  the  total  number  of  ob- 
servations; and  S  indicates  the  operation  of  summing 
all  the  terms  contained  within  the  parentheses. 
To  facilitate  the  following  development,  let  us  put 

^  ^  N  N 

Since  y  =  mx  -\-h,  substitute  this  value  of  ?/  in  (1) .    Then 

V      i:\nAmx +h  —  y,y} 


(2) 


N  N 

1,{n^{m^x~  +  2mbx  +  6-  —  2mxy^  —  25^^  +  yl)  ] 

~  N 


Now  — ^ —  =  o'x  +  ^'j  where  x  =  the  mean  of  the  x's, 

and  (Tx,  the  standard  deviation  of  the  x's.    This  follows 

2(72,  x) 
from  Theorem  III ;  — ■—-  =  x,  by  the  definition  of  the 

arithmetical  mean;  — ^   =  —  =  1; 


The  Mathematics  of  Correlation  35 

^  ""  J^""'  =  the  mean  of  the  xy  products.     This  may  be 
seen  to  be  true  from  the  following: 


Vx, 


Consequently, 

But  what  is  true  of  this  particular  array  is  true  of 
all  the  arrays  and  since  there  are  A^  products  xy,  the 

y^  (71  iril   1 

value  of        ^        is  the  mean  of  the  xy  products.    Let 

us  call  the  mean  of  the  xy  products  p^y.    Continuing  the 

investigation  of  equation  (3) ,  we  have,  — ^r-^  =  the 

mean  of  the  several  values  of  ?/^  =  y,  where  y  is  the  mean 

of  all  the  ordinates. 

S(n  if) 

— ^r-^  =  (the  standard  deviation  of  the  ^^'s)'-  +  y''- 

This  follows  from  Theorem  III.    Let  us  put  a-y^  for  the 
standard  deviation  of  the  yj^. 

If  now  we  make  the  proper  substitutions,  we  may 
write  equation  (3)  as  follows : 

V 

(4)         ^  =  m\al  +  x')  +  2mhx  +  h"- -  2mp^y 

-2by+al+r 


Our  problem  is  to  find  the  values  of  m  and  b  that  will 

make  — r  a  minimum. 

N 

Let  Ci,  62  be  two  quantities  so  small  that,  for  the  pur- 


36       Forecasting  the  Yield  and  the  Price  of  Cotton 

pose  in  hand,  we  may  neglect  their  squares  and  products 
and  write 

(m  +  e,y  =  m'-+  2mei;  (6  +  62)-  =  b'~  +  2662; 
(w  +  ei)  {b  +  62)  =  mb  +  ezm  +  eib. 

Suppose  that,  when  in  equation  (4)  we  put  (m  +  eO 
for  m  and  (6  +  62)  for  6,  we  get 

(5)  ^  =  (m  +  ei)-^  (o-'i  +  X')  +  2(m  +  ei)  (b  +  62)^ 

+  (6  +  62)-  -  2(m  +  ei)  p,,  -  2(6  +  62)  ij  +  <  +  ^'; 
=  (m2  +  2mei)  ((r|  +  x^)  +  2(m6  +  egm  +  ei6)  x  + 
(62  +  2662)  - 2(m  +  ei)  p.,  -  2  (6  +  62)  y  +  crf^  +  2/^. 

Let  us  consider  a  httle  more  concretely  the  meaning 
of  equations  (4)  and  (5).  Equation  (4)  indicates  that 
if,  in  the  straight  line  y  =  mx  +  b,  we  assign  any  given 
values  to  m  and  b,  then  the  mean  square  of  the  devia- 
tions of  the  points  from  the  straight  line  is  equal  to  — . 

Equation  (5)  indicates  that  if  we  change  the  value  of  m 
in  (4)  to  (m  +  ei)  and  the  value  of  b  in  (4)  to  (6  +  62) 
where  Ci  and  62  are  very  small  quantities,  then  the  mean 
square  of  the  deviations  of  the  points  from  the  straight 

line  y  =  {m  +  ei)  X  -i-  (6  +  62)  is  given  by  — .    If  m  and 

V .       .  y 

b  in  (4)  are  so  determined  that  t^  is  a  minimum,  then  — 

V 

is  greater  than — ,  and,  by  subtracting  (4)  from  (5) ,  we  have 

V'-V 

(6)  =  2mei  {al  +  x'-)  +  2(^2^  +  ej))  x  +  26^2 

-2eip^y-2e2y; 
=  2ei  {m{(Tl  +  ^^)  +  bx~P:,y]  +  2e2  {mx  -\-  b  —  y}. 


The  Mathematics  of  Correlation  37 

V 

But  when  m  and  h  are  so  determined  as  to  render  — 

N 

V         V 
a  minimum,  and  ei  and  62  are  very  small,  —  and  T7  are, 

for  practical  purposes,  equal  and  equation  (6)  may  be 
put  into  the  form 

(7)  2ei{m(o--  +  a;-)  +  hx-v.y}  +  2e2{  mx  +  h-y]  =  0. 

In  order  for  equation  (7)  to  be  a  true  equation,  a  suffi- 
cient condition  is  that  the  coefficients  of  2ei  and  2^2 
shall  each  be  zero ;  that  is 

(i)  m{(Tl  +  X-)  +  bx-p^y  =  0; 
(ii)  7nx  -{-  b~y  ^  0. 

Solve  these  two  equations  for  m  and  b.  Multiply  (ii) 
by   X   and   subtract    the   result    from    (i).      We   get 

mal  —  p^y  -\-xy  =0,  and,  consequently,  m  =  -^^^-^ — . 

Substitute  this  value  of  m  in  (ii)  and  solve  the  resulting 

T)    —  xii 
equation  for  b.     We  obtain  b  =  y—        ^ — x.      If  we 

substitute  these  values  of  m  and  b  in  the  equation  to 
the  straight  line,  y  =  mx  +  6,  we  get 


(8)  2/-^^^    .^'^x  + 


v^y  —  xy  - 
y —. —  X 


V 
This  is  the  equation  to  the  straight  hue  that  makes  — , 

the  mean  square  deviation  of  the  points  from  the  line,  a 
minimum;  it  is  the  straight  line  that  fits  best  the  data. 

If  our  sole  purpose  were  to  find  the  line  fitting  best 
any  given  data,  we  might  stop  here.    We  should  then 


319341 


38       Forecasting  the  Yield  and  the  Price  of  Cotton 

compute  from  the  given  data  the  values  of  the  constants 
in  equation  (8),  and,  by  substituting  these  values  in 
that  equation,  obtain  the  equation  in  its  numerical 
form.  Our  problem,  however,  is  not  completely  solved 
by  finding  the  equation  connecting  the  two  variables 
X  and  y.  We  wish  to  know,  in  any  given  case,  how  closely 
the  two  variables  are  associated.  In  the  particular 
case  which  we  have  taken  to  illustrate  our  mathemat- 
ical methods,  we  wish  not  only  to  know  the  equation 
connecting  the  fluctuations  of  New  York  futures  from 
their  general  trend  with  the  fluctuations  of  the  aver- 
age prices  of  spot  cotton  from  their  general  trend,  but 
we  wish  to  know  how  closely  the  prices  of  futures  and 
the  prices  of  spot  cotton  are  connected.  To  approach 
this  last  problem  we  simplify  equation  (8) . 

We  have  agreed  to  call  x  the  mean  of  the  x's  in  the 
scatter  diagram,  and  y,  the  mean  of  the  i/'s.  Suppose 
we  call  the  point  in  the  scatter  diagram  whose  coor- 
dinates are  {x,  y)  the  mean  of  the  system  of  points, 
and  inquire  whether  the  straight  line  described  by  equa- 
tion (8)  passes  through  the  mean  of  the  system  of  points. 
If  the  line  passes  through  this  point,  the  coordinates 
of  the  point  {x,  y)  must  satisfy  the  equation.  Substitute 
X,  y  respectively  for  x  and  y  in  (8) .    We  obtain 

(9)         y  =  ^ x+  \y ^—^ ^  ,  or, 

Vxy  —  xy  _       _       p,y  —  xy  _ 
V ^^^  =  !/-^^x. 

But  this  is  a  true  and  identical  equation,  and  conse- 
quently the  line  described  by  equation  (8)  passes 
through  the  mean  of  the  system  of  points  on  the  scatter 


The  Mathematics  of  Correlation  39 

diagram,  that  is  to  say,  the  point  whose  coordinates 
are  {x,  ij). 

The  fact  that  the  best-fitting  hne  passes  through  the 
point  (:c,  y)  enables  us  to  simplify  equation  (8).  By 
transposing  we  may  write  (8)  as  follows: 

(10)  {y-y)  =  ^'^  ~  ^^  {X  -  X). 

The  quantity  {y  —  y)  is  the  deviation  of  the  ordinate 
of  the  best-fitting  straight  line  from  the  mean  of  the 
y's,  and  may  be  represented  by  Y;  the  quantity  (x  —  x) 
is  the  deviation  of  the  abscissa  of  the  line  from  the 
mean  of  the  x's,  and  may  be  represented  by  X.  Since, 
as  we  have  just  proved,  the  line  passes  through  the  point 
(x,  y),  if  we  transfer  the  origin  from  zero  to  the  point 
{x,  y),  equation  (10)  may  be  written 

(11)  F  =  ^"^  T  ^^  X. 

The  effect  of  transferring  the  origin  to  the  point  {x,  y) 
is  to  get  rid  of  the  value  of  h  in  the  equation  to  the 
straight  line. 

10    — '  xfi 
We  shall  now  examine  the  quantity  -^^— ^ =  ^i, 

which  appears  in  both  (10)  and  (11).  We  know  that 
p^y  is  the  mean  value  of  the  products  xy.  Let  us 
define  a  new  quantity  tr^y  to  be  the  mean  product  of 
the  deviations  of  x  and  y  from  their  respective  means. 
Then,  by  definition, 

_'L{^^-x){y~J)_  _i:{xy)        2(1/)       -2(x) 


40       Forecasting  the  Yield  and  the  Price  of  Cotton 

-DUi  —  Pjj/^     N    ~         N    ~  ^' 

Therefore   ir^y  =  p^y  —  xy,    and    we    may   write   m  = 
^^   , — -  =  — v-     Make  this  substitution  in  equations 
(10)  and  (11)  and  we  get 

(12)  {y~y)-'^ix~x)', 

(13)  F  =  ^X. 

If,  as  a  further  step,  we  define  r  to  be  a  quantity  such 


that  r  =  -^^,  then,  by  substituting  in  (12)  and  (13), 

we  may  write  the  equation  to  the  best-fitting  straight 
hne  in  either  of  the  following  forms : 

(14)  (y  —  y)  =  r-^ix-x). 

(15)  Y  =  r^X. 

CTx 

The  quantity  r  in  these  equations  is  called  the  coefficient 
of  correlation. 


The  Coefficient  of  Correlation 

An  inspection  of  equations  (14)  and  (15)  shows  that 
in  order  to  secure  the  best  fit  of  a  straight  line  to  given 
data,  all  that  is  necessary  is  to  compute  froin  the  data 
the  values  of  x,  y,  <t^,  ay,  r,  and  to  make  the  proper 
substitutions  in  (14)  and  (15).  We  have  already  dis- 
cussed methods  of  computing  x,  y,  o-j.,  ay,  and  we  now 


The  Mathematics  of  Correlation  41 

reach  the  question  of  the  best  method  of  computing 

r.     As  we  have  just  seen,  r  =  — -  =  -^ — , 

and  if  we  were  indifferent  to  the  labor  of  computation, 
we  might  use  this  formula  to  ascertain,  in  any  concrete 
case,  the  value  of  r.  We,  however,  found  methods  for 
computing  x,  y,  a,.,  ay  by  working  with  deviations 
from  arbitrary  quantities  as  origins,  and  we  now  pro- 
ceed to  develop  a  method  of  computing  r  by  retaining 
the  same  arbitrary  origins  which  we  used  in  calculating 
the  means  and  the  standard  deviations.     We  wish  to 

find  '^ -^—^ '—  and  to  derive  its  value  by  working 

with  deviations  of  the  x's  and  i/'s  from  arbitrary  origins. 

Theorem  IV.  The  mean  product  of  the  deviations  of 
two  correlated  variables  from  their  respective  arithmetical 
means  is  equal  to  the  mean  product  of  the  deviations  of  the 
two  variables  from  arbitrary  origins,  minus  the  difference 
between  the  arbitrary  origin  and  the  mean  of  the  one  vari- 
able multiplied  by  the  difference  between  the  arbitrary 
origin  and  the  7nean  of  the  second  variable. 

Let  the  observations  be  (xi,  yi);  (xz,  2/2);  {xs,  y^)  .  .  . 
(x,„  y,).  Let  P  be  the  arbitrary  origin  from  which  we 
measure  the  deviations  of  the  .r's,  and  Q  be  the  arbitrary 
origin  from  which  we  measure  the  deviations  of  the  |/'s. 
Let  the  deviations  of  the  x's  from  P  be  represented  by  x' 
and  the  deviation  of  the  ^'s  from  Q  be  represented  by  y' . 
Let  the  deviations  of  the  .r's  from  x  be  represented  by  X, 
and  the  deviations  of  the  |/'s  from  ij  be  represented  by 
F.  If  we  put  x  —  P=  d^,  and  y  ~Q  =  d,,,  our  Theorem 
IV  is  that 

i:{x-x){:y-y)  ^  I:(.tV) 
N  N 


42       Forecasting  the  Yield  and  the  Price  of  Cotton 

We  have 

(16) 

x[  =  Xi  ~  P  =  Xi  —  {x  ~  d^)  =  (xi  -  x)  +  dj.  =  Xi+  d^, 

Xo  =  X2  -  P  =  X2   -  {X  -  4)  =   (X2  -  x)  +  4  =  ^2  +  4, 


x'„  =  x„  ~  P  =  x„  -  (x  -  dj  =  {x„  -  x)  +  4  =  X„+  d^. 
Similarly, 

y'i  =  yi-  Q  =  yi-  (y  -  dy)  =  {yi  ~  ij)  +  dy=  Fi  +  dy, 
y'o-  y2~Q  =  y2  -  {y  -  dy)  =  {y^  -  y)  +  dy=  Yo  +  dy, 


y'n  =  yn-  Q  =  yn  -  (y  -  dy)  =  (?/„  -  y)  +  dy=  F„+  dy. 

Therefore, 

x[y[  =  (Xi+  4)(Fi  +  dy)  =  XiFi  +  dyXi  +  d,Y^+dJy, 
x',y[  =  (X2+  d,){Y.  +  dy)  =  X2F2  +  c^^Xo  +  d,Yo+dJy, 


xWn=^{X„+  4)(F„+  d,)  =  X„F„+  4X„+  d,F„+  4d,. 
Summing  both  sides  of  the  equation,  we  get 

S(xY)  =  2(XF)  +  dyZiX)  +  dMY)  +  Ndjy. 

But,    according    to    Theorem    I,    S(X)  =  S(F)  =  0, 
and,  consequently, 

S(xV)  =S(XF)  +X4rf„or, 

,^_.     S(XF)      2(x-x)(7/-7/)      S(a;V)        ,  , 
(17)     -^^  = ^^ =  -^ djy. 


This  formula  gives  us  a  method  of  computing  the 

the  value  of  r,  t] 

i:{x  —  x){y  -  y) 


factor  — ^ — ^  in  the  value  of  r,  the  formula 


for  which  we  know  is,  r  =- 


N(r^(Ty 


1         1 

1 

^ 

m 

o 

"^ 

CN 

-*< 

^ 

N 

X 

-f 

»o 

o 

•— 1 

CO 

CO 

LO 

o 

C-l 

o 

--0 

LO 

M 

t^ 

0 

(N 

1 

H 

1 

»0        lO 

+  + 

LO         "O 

ri  ■"  c^ 

-- 

LO 

+ 

O   ^ul 

LO 

o  Oro 

~ -<3~ 

(N 

+     + 

+ 

^     -  L'^ 

1 

I--    C  Q 

•* 

m 

o  ^-^ 

^3C^S' 

^O-hS" 

==^S 

(N^C? 

lO^^S 

t^ 

^ 

+  + 

' 

^"^ 

^^ 

^^ 

^^ 

+ 

-B 

lO 

1 

'             '             1 

'             I 

o 

.o     >o 

^^ 

O 

c 

o*"o 

1    ^    1 

^^  -)■  x~ 

^t^?5 

.__  ^x 

X  —  00 

O 

o 

e 
> 

+  + 

1         X 

+ 

o  _o 

^H    0  r}4 

LO 

1 

c  *'o 

1.0           LO 

1   ^    1 

1    '-''    1 

o  ^  3 

^oS" 

^-1 

cc  o^ 

LO 

o 

0 

+  + 

1        J^ 

1         J^ 

^^ 

+ 

ti 

a 

^^ 

■- 

O   jjiO 

CI 

o 

o  ^  o 

t^ 

o 

S 

1   + 

O-H  o 

c-oc 

=  C^'S 

—  c5S 

Ot^O 

o 

1 

V- 

o 

a 

:5ol2 

LO 

l'      l' 

Tf^'^ 

-o  —  S" 

Nl-2 

LOiO 

o^g 

I^J 

o 

f 

ci 

1           1 

1 

t-^ 

> 

: 1 

oj 

O 

9    9 

x.-<g 

OTf§' 

^ooi 

=^--=S 

S  -rS 

i^ 

1 

o 

1      1 

'^ ' 

^— ' 

^^ 

1 

« 

I 

___ 

o 

O   _iO 

^ 

Oi 

^ 

-^-o 

'•^i 

„,^ 

^?       CC 

o 

u^ 

1           1 

'^^3 

1  "  1 

•t 

I 

1     1 

1     1 

1 

lO  .o 

ro  2o 

LO 

2^§ 

,-^ 

,■ 

Tjl  ^^ 

n 

1 

1     1 

1 

lO   -lO 



o  Om 

LO 

1    1 

2-^2 

ic  — 1 C? 

m 

1 

lO  -L-: 

*-0   _  »0  '  lO    ^  lit 

1-^        u- 

LO     ,"0 

.0    -T 

O   _u- 

LO  _ir 

K5      ir 

lO  -ir 

lO   _iO 

w  Qc^ 

(N  Oh-  ir^  Oo 

CNJ  3  "^^ 

fc:  2  22 

M    OCM 

■N  On 

t^5<^ 

oq  °t- 

t^  5=^ 

c^  Si^ 

c^  *^c^ 

c^  *^^    ^^^ 

o""c 

o  ■*^c 

O'*^'- 

-H-^C^ 

C>1  ■*^IN 

tn 

tn 

1        1 

1    1  1    1 

1       1 

1       1 

1  ^ 

-+    4 

-+    4 

-+    4 

-+    4 

-   +      + 

"5 

a 

o 

s.^tip  g  JO  snuam  aAissajaoicI  aq^  mojj  saintinj  ?{joj^  .aa^y  jo  suoi^Bn^oni  j[ 

44       Forecasting  the  Yield  and  the  Price  of  Cotton 

The  data  in  Table  4  will  serve  to  illustrate  the  method 

of  computing  the  coefficient  of  correlation.^    We  have 

1  ,,    ,        2i(x  —  x){y  —  y)        ,  ,,    ,  l!t{x—x){y  —  y) 

proved  that  r  =^ ^-^ — ^,  and  that  -^ ^-^ — — 

Na^dy         '  N 

'Lix'ii') 
=  — d^dy]    and    we    recall    that    d,.  =  x  --  P, 

N 

where  P  is  the  arbitrary  origin  from  which  x'  is 
measured,  and  dy  =  y  —  Q  where  Q  is  the  arbitrary 
origin  from  which  y'  is  measured.  In  the  correlation 
table  which  is  given  in  Table  4,  P  is  the  origin  from 
which  are  measured  the  fluctuations  of  the  average 
spots  about  the  progressive  means  of  5  days,  and  is 
taken  at  the  point  zero,  which  lies  mid-way  between 
(-  .015)  and  (+  .015).  The  values  of  x',  the  fluctua- 
tions of  average  spots,  are  the  distances  to  the  right 
and  to  the  left  of  the  arbitrary  origin,  and  the  sign  of  x' 
is  positive  or  negative  according  as  the  distance  is  to 
the  right  or  to  the  left  of  the  arbitrary  origin.  In  a 
similar  manner,  the  arbitrary  origin  Q,  from  which 
are  measured  the  fluctuations  of  New  York  futures 
about  the  progressive  means  of  5  days,  is  taken  at  the 
point  zero  which  lies  mid-way  between  ( —  .025)  and 
(+  .025).  The  fluctuations  from  Q,  which  are  desig- 
nated by  y',  are  negative  toward  the  upper  end  of  the 
table  and  positive  toward  the  lower  end.  Just  as  in  the 
scatter  diagram,  which  is  given  in  Figure  4,  the  275 
observations  were  represented,  according  to  their  co- 

1  The  method  described  in  the  text  is  the  one  most  frequently  required 
in  actual  experience.  Where,  however,  the  number  of  observations  is 
small,  which  happens  to  be  the  case  with  a  large  part  of  the  data  in  this 
Essay,  a  slight  alteration  of  the  procedure  described  in  the  text  is  neces- 
sary. A  complete  illustration  of  the  method  of  correlation  when  the 
observations  are  few  in  number  is  given  in  Chapter  III,  Table  6. 


The  Mathematics  of  Correlation  45 

ordinates,  by  points  on  the  diagram,  so  in  the  correla- 
tion table  each  observation  falls  in  some  one  of  the  cells 
composing  the  Table.  The  figure  in  the  middle  of  the 
cell  gives  the  number  of  observations  in  the  cell;  for 
example,  in  the  upper  left-hand  corner  there  is  one  ob- 
servation, which  means  that  out  of  275  days  observa- 
tion, there  was  one  day  when  the  fluctuation  of  average 
spots  was  between  ( —  .165)  and  ( —  .135)  from  the  gen- 
eral trend  of  spots,  and  the  fluctuation  of  New  York 
futures  was  between  (—  .275)  and  (—  .225)  from  the 
general  trend  of  New  York  futures.  In  the  same  cell 
in  the  upper  left-hand  corner  of  the  correlation  table 
there  is  above  the  figure  1  the  figure  25,  and  below  the 
figure  1,  the  figure  (25).  A  similar  arrangement  is 
followed  in  all  of  the  cells  in  which  observations  occur, 
and  we  now  proceed  to  explain  its  meaning.  The  work- 
ing unit  in  the  classification  of  the  fluctuations  of  spots 
is  .03,  and  in  the  classification  of  the  fluctuations  of 
New  York  futures,  it  is  .05.  The  range  of  the  fluctua- 
tions of  spots  is  from  the  mid-value  of  the  first  cell 
on  the  left  to  the  mid-value  of  the  last  cell  on  the  right, 
that  is,  from  (—  .15)  to  (+  .15),  or,  since  the  working 
unit  of  the  x"s  is  .03,  the  range  is  from  (—5)  to  (-|-  5) 
working  units.  Similarly,  the  range  of  the  ?/"s  is  from 
(—  .25)  to  (+  .25),  or,  since  the  working  unit  is  .05,  the 
range  is  from  (—  5)  to  (+  5)  working  units. 

Returning  now  to  the  one  observation  in  the  upper 
left-hand  corner  of  the  correlation  table,  we  find  that 
its  distance  from  the  zero  point  of  the  .t"s  is  (—  5)  work- 
ing units,  and  from  the  zero  point  of  the  ^"s  is  also 
(—5)  working  units.  The  product  of  these  two, 
which  is  x'y',  is  ( —  5)  ( —  5)  =  25,  and  this  explains 


46       Forecasting  the  Yield  and  the  Price  of  Cotton 

the  figure  25  at  the  top  of  this  one  cell.  Since  there  is 
only  one  observation  in  this  cell,  if  we  weight  the  prod- 
uct 25  by  1  we  get  (25),  which  explains  the  figure 
(25)  at  the  bottom  of  this  particular  cell.  To  summar- 
ize, the  figure  in  the  middle  of  the  cell  is  the  frequency 
of  the  observations;  the  figure  at  the  top  of  the  cell  is 
the  product  x'y'  in  working  units;  and  the  figure  at  the 
bottom  of  the  cell  is  x'y'  weighted  according  to  the 
number  of  observations  in  the  cell.  The  heavy  lines 
that  pass  from  the  top  to  the  bottom,  and  from  the 
left  to  the  right  of  the  correlation  table  divide  the  latter 
into  four  large  divisions.  All  of  the  products  in  the 
cells  of  the  upper  left-hand  and  lower  right-hand  divi- 
sions are  positive,  and  all  of  the  products  of  the  other 
two  divisions  are  negative.  If  we  sum  all  of  the  positive 
products  separately  and  then  all  of  the  negative  prod- 
ucts, their  difference  will  give  us  Ii{x'y');  and  if 
we  then  divide  this  result   by  275  we  shall  obtain 


S(^'2/') 


If  we    indicate    by   S(+  x'y')    the   sum   of 


N 

the  positive  products,  and  by  2(— .x'l/')  the  sum  of 
the  negative  products,  we  find  from  Table  4  that 

S(+  x'y')  =  (25)  +  (15)  +  (5)  +  (32)  +  (4)  +  (18)  + 
(6)  +  (8)  +  (24)  +  (32)  +  (12)  +  (4)  +  (18)  +  (14)  -|- 
(25)  +  (19)  4-  (28)  +  (18)  +  (8)  +  (28)  +  (18)  +  (8)  + 
(6)  +  (6)  +  (18)  +  (12)  +  (15)  +  (16)  +  (20)  +  (25)  = 
487;  and  2:(-  x'y')  =  (-  3)  +  (-  4)  +  (-  5)  + 
(_  5)  +  (_  2)  =  -  19.    Consequently,  :2(x'y')  =  487  - 

19  =  468,   and   5^^  =  ^=1.7018.     The   quantity 

-           .             ,    .    2(.T  —  x){y  —  y) 
that  we  wish  to  deternune  next  is  — , 


The  Mathematics  of  Correlation  47 

'Z(x'v') 
which    we    know    is    equal    to    — — d_,d,^.      We 

have  found  in  the  early  part  of  this  chapter  that 
d^  =  —  .073  in  working  units;  and  just  as  we  deter- 
mined d^  we  can,  in  a  similar  manner,  determine  dy.  The 
actual  computation  show^s  that  dy  =  —  ,026  in  working 
units.     Consequently,  dj^  =  (-  .073)  (—  .026)  =  .0019, 

and  5^^  -  d,d,  =  1.7018  -  .0019  =  1.6999,  which  is, 

therefore,  the  value  of — .    But  the  coeffi- 

,  ,•         •          1  ,    '^{^  —  ^)iv  —  v) 
cient  of  correlation  r  is  equal  to — ,  and 

since,  in  working  units,  a^.  =  1.405  and  a^  =  1.694,  we 

1.6999 

have  r  = =  .714. 

2.3801 

Only  one  other  short  step  is  needed  to  get  the  equa- 
tion to  the  straight  line  that  fits  best  the  data.  (See 
Figure  5.)  We  know  that  the  equation  to  the  best- 
fitting  straight  line  is   {y  —  y)  =  r—  {x  —  x),  and  all 

that  is  necessary  is  to  substitute  for  the  symbols 
their  numerical  values:  x  =  —  .002;  y  =  —  .001, 
0-,  =  (1.405)  (.03)  =  .042;  o-„  =  (1.694)  (.05)  =  .085; 
r  =  .714.  The  proper  substitution  gives  for  the  equa- 
tion to  the  line,^  y  =  lA5x  +  .002. 

The  equation  y  =  1.45a:  -j-  .002  gives  the  law  of  the 
association  of  the  price  of  New  York  futures  with  the 
price  of  average  spots  in  the  South.  Whatever  may  be 
the  value  of  x,  which  is  the  fluctuation  of  average  spots 

1  The  slight  difference  between  this  equation  and  the  one  given  in 
Figure  5  is  due  to  the  fact  that  in  computing  the  latter  equation  Shep- 
pard's  correction  was  used  in  getting  tlie  values  of  cr^  and  cr,/. 


48       Forecasting  the  Yield  and  the  Price  of  Cotton 

from  their  general  trend,  the  above  equation  enables 
one  to  compute  the  most  probable  value  of  y,  which 
is  the  fluctuation  of  New  York  futures  from  their 
general  trend.  For  example,  if  x  should  be  equal  to 
(+  .15),  the  most  probable  value  of  y  is,  y  =  1.45(.15) 
+  .002  =  .22. 

The  geometrical  significance  of  r.    We  have  proved 
that  the  equation  to  the  best-fitting  straight  line  is 

Y  =  r  —  X.     Suppose  we  write  this  equation  in  the 

following  form : 

This  expression  enables  us  to  form  a  picture  of  the 
geometrical  significance  of  r.  Equation  (18)  shows  that 
if  the  X's  are  expressed  in  terms  of  a_,  and  the  F's  in 
terms  of  ay,  then  r  is  the  tangent  of  the  angle  which 

the  straight  line  makes  with  the  axis  of  (^ )  •  The  value 
of  r  shows  the  proportional  change  in  (^  j  correspond- 
ing to  a  unit  change  in  (^). 

The  limits  to  the  value  of  r  are  +  1.     The  equation 

to  the  best-fitting  straight  line  is  Y  =-  r  —  X.     Let  the 

iV  observations  be  (Xi,  Yi);  (X2,  F2) ;  .  .  .  (X„,  FJ. 
Then,  by  the  definition  of  V,  we  have 

V={Y^-r^Xry  +  iY,-Ax,y+.  .  .  +  {Y,-r^X,y, 
-2(F2)-2r-'S(XF)+r--f2(X'-^). 


(19)  -  =  (7^(1  -  r^). 


The  Mathematics  of  Correlation  49 

But  S(F2)  =  AV^;S(X^)  =  AV-  S(XF)  =  .Yrcr.cr,. 

Consequently, 

V  =  Nal~  2Nr'(Tl  +  Nr^a^  =  No^il  -  f'),  and  there- 
fore, 

V 

N 

V 
But  — ,  being  the  mean  square  deviation  of  the  observa- 

JS 

tions  from  the  straight  hue,  is  a  positive  quantity. 
Therefore,  r  cannot  exceed  ( +  1)  nor  be  less  than  (—1). 
The  fact  which  was  brought  out  a  moment  ago,  namely, 
that  r  is  the  tangent  of  the  angle  which  one  straight 
line  makes  with  another,  shows  that  the  value  of  r  may 
be  positive  or  negative  according  to  the  inclination  of 
the  line. 

The  use  of  r  in  the  problem  of  forecasting.  Equation 
(19)  gives  us  a  formula  which  is  of  the  very  first  impor- 
tance in  our  effort  to  forecast  economic  events.     We 

y 

have  —  =  o-;(l  —  r'),  and  this  is  the  measure  of  the 

mean  square  deviation  of  the  points  from  the  straight 
line  that  fits  best  the  observations.     WTien  r  =  (+  1) 

V 

or  (—  1),  —  equals  zero;  all  of  the  points  lie  on  the 

A 

straight  line;  and,  by  means  of  the  equation  to  the 
straight  line,  we  can  predict  exactly  the  value  of  y  cor- 
responding to  a  given  value  of  x.  But  it  is  very  seldom 
that  r  =  +1,  and  when  r  lies  between  these  two  limit- 
ing values,  we  can  still  forecast  results  with  a  knowledge 
of  the  probabihties  in  favor  of  the  forecast.     Let  us 

put  aS  =  \  /   {^^   =  o-</  v/l  -  r-.     We  know  from  the 


50       Forecasting  the  Yield  and  the  Price  of  Cotton 

Table  of  the  Probability  Integral  that  when  the  distribu- 
tion of  the  points  about  the  straight  hne  is  normal,  99.7 
out  of  100  observations  he  within  a  deviation  from  the 
straight  hne  equal  to  +  3aS;  95  out  of  100  lie  between 
+  2aS;  and  68  out  of  100  lie  between  +  S.  The  equa- 
tion to  the  best-fitting  straight  line  enables  us  to  com- 
pute the  most  probable  value  of  y  corresponding  to  a 
given  value  of  x;  the  value  of  S  enables  us  to  say  within 
what  limits  any  proportion  of  the  actual  observations 
are  scattered  about  the  straight  line.  The  coefficient 
of  correlation  r  is  the  coefficient  which  we  have  been 
seeking  as  a  measure  of  the  degree  of  association  be- 
tween two  variables.  Where  the  association  between 
the  variables  is  perfect,  r  =  +  1,  >S  =  o-^/^l  —  r-  =  0, 
and  from  the  knowledge  of  the  one  variable  we  can, 
by  means  of  the  equation  to  the  best-fitting  straight 
line,  forecast  the  other  variable  with  perfect  accuracy. 
When  the  association  between  the  two  variables  is  not 
perfect,  r  falls  between  the  limiting  values  +  1,  and 
S  =  o-^v  1  —  r-  shows  the  accuracy  with  which,  using 
the  equation  to  the  best-fitting  straight  line,  the  mag- 
nitude of  the  one  variable  may  be  predicted  from  a 
knowledge  of  the  other. 

We  may  illustrate  these  points  by  the  problem  of  the 
relation  between  New  York  futures  and  average  spot 
values  in  the  South.  We  have  found  that  the  best- 
fitting  straight  line  connecting  the  fluctuations  in  New 
York  futures  with  the  fluctuations  in  the  price  of  spot 
cotton  is  ^  =  1.45x  +  .002.  For  any  given  value  of  x, 
representing  the  fluctuation  in  the  price  of  spot  cotton, 
we  can  predict,  by  means  of  this  formula,  the  most 
probable  fluctuation  in  the  price  of  New  York  futures. 


The  Matheviatics  of  Correlation  51 

We  are,  however,  not  content  to  forecast  the  most 
probable  values  of  y,  but  we  wish  to  know,  in  addition, 
the  degree  of  accuracy  of  the  forecasts.  The  formula 
that  has  just  been  developed  supplies  an  answer  to  this 
latter  question.  Since  r  =  .714  and  a-y  =  .085,  there- 
fore *S  =  (7^  V  1  —  r-  =  .06,  and  from  what  we  have 
learned  about  the  significance  of  S,  we  know  that,  when 
we  use  the  formula  y  =  lA5x  +  .002  as  a  prediction 
formula,  in  99.7  per  cent  of  all  the  forecasts  the  error 
will  be  less  than  +  3S;  in  95  per  cent  of  all  the  fore- 
casts, the  error  will  be  less  than  +  2*S;  and  in  68  per 
cent  of  all  the  forecasts,  the  error  will  be  less  than  +  S. 

In  beginning  this  chapter  we  referred  to  the  official 
statement  that  "since  the  cotton  futures  Act  went  into 
operation,  future  quotations  have  fairly  reflected  spot 
values  in  both  New  York  and  New  Orleans,  and  also  in 
a  general  way  over  the  entire  South."     We  made  the 
comment:   ''Just  what  is  meant  by  fairly?    How  can 
one  measure  the  degree  of  association  between  futures 
and  spot  values?     Or,  to  put  the  question  in  another 
form,  suppose  one  knew  the  probable  spot  values  in  the 
South,  how  could  one  forecast  the  price  of  futures"  on 
the  cotton  exchange  at  New  York?    All  of  these  ques- 
tions may  now  be  answered  in  a  definite,  numerical 
way:  the  degree  of  association  between  futures  in  New 
York  and  spot  values  in  the  South  is  measured  by 
r  =  .714 ;  the  formula  by  which  futures  may  be  predicted 
from  the  knowledge  of  spot  values  is  ?/  =  1 .45a:  -|-  .002 ; 
and  the  error  of  the  forecasts  by  means  of  this  formula 
is  measured  by  *S  =  .06. 


CHAPTER  III 

THE  GOVERNMENT  CROP  REPORTS 

"The  consequences  of  false  reports  concerning  the  condition  and 
prospective  yield  of  the  cotton  crop  alone  maj'  be  very  damaging.  If 
there  were  no  adequate  Government  crop-reporting  service,  and  by 
misleading  reports  speculators  should  depress  the  price  a  single  cent  per 
pound,  growers  would  lose  $60,000,000  or  more;  if  prices  were  improp- 
erly increased,  the  manufacturers  and  allied  interests  would  be  affected 
to  a  proportionate  degree." 

— Circular  17,  Bureau  of  Statistics,  U.  S.  Department  of  Agriculture. 

The  character  and  the  aim  of  the  official  crop-report- 
ing service,  the  definition  and  the  use  of  technical 
terms,  and  the  actual  procedure  in  crop-forecasting 
have  been  described  in  publications  of  the  Department 
of  Agriculture.^  The  official  documents  might,  of 
course,  be  summarized,  but  it  seems  advisable  to  quote 
in  full  those  statements  that  have  a  bearing  upon  the 
subject  of  the  present  and  subsequent  chapters  of  this 
Essay. 

The  Character  and  the  Aim  of  the  Crop-Reporting  Service 

The  Department  of  Agriculture  "is  said  to  have  been  conceived 
in  the  far-sighted  wisdom  of  Washington,  who,  as  President,  sug- 
gested the  organization  of  a  branch  of  the  National  Government  to 
care  for  the  interests  of  the  farmers;  and,  in  the  practical  activity  of 
Franklin,  who,  as  agent  of  Pennsylvania  in  England  sent  home  silk- 
womi  eggs  and  mulberry  cuttings  to  start  silk  growing.    But  the 

^  I  wish  to  acknowledge  with  hearty  thanks  the  courteous  helpful- 
ness of  the  officials  of  the  Department  of  Agriculture  in  supplying  me 
with  statistical  material.  Mr.  Charles  J.  Brand,  Mr.  Leon  M.  Ester- 
brook,  Mr.  George  K.  Holmes,  and  Mr.  Nat  C.  Murray  were  particu- 
larly generous  in  their  assistance. 


The  Government  Crop  Reports  53 

conception  did  not  materialize  into  form  mitil  1839,  when,  on  the 
recommendation  of  the  Hon.  Henry  L.  Ellsworth,  Commissioner  of 
Patents,  an  appropriation  of  $1,000  was  made  by  Congress  for  the 
'collection  of  agricultural  statistics,  investigation  promoting  agri- 
cultural and  rural  economy,  and  the  procurement  of  cuttings  and 
seeds  for  gratuitous  distribution  among  farmers.' 

"An  agricultural  section  was  established  in  the  Patent  Office,  and 
the  collection  of  seeds  and  the  publication  of  agricultural  statistics 
and  scientific  articles  on  agricultural  topics  were  placed  directly 
under  control  of  the  Commissioner  of  Patents,  at  that  time  an 
official  of  the  Department  of  State;  the  work  continued  under 
succeeding  Commissioners  of  Patents  until  1849,  when  the  Depart- 
ment of  the  Interior  was  established,  and  the  Patent  Office,  with  its 
agricultural  section,  became  a  part  of  it.  From  that  time  until 
1862,  when  the  section  was  made  a  separate  Department  under  a 
Commissioner  of  Agriculture,  the  agricultural  work  was  done  by  the 
chief  of  the  section  of  agriculture  in  the  Patent  Office,  under  the 
direction  of  the  Commissioner  of  Patents. 

"From  1862  until  1889,  when  the  Department  was  raised  to  the 
dignity  of  a  cabinet  office,  the  work  was  prosecuted  under  the  Com- 
missioner of  Agriculture,  independently  of  the  Department  of  the 
Interior. 

"Under  President  Cleveland's  first  administration  the  Depart- 
ment became,  on  February  11,  1889,  one  of  the  Executive  Depart- 
ments of  the  Govermnent."  ^ 

"The  first  enactment  authorizing  the  collection  of  agricultural 
statistics  ^  by  the  Department  of  Agriculture  was  the  act,  passed 
May  15,  1862,  establishing  the  Department,  'the  general  design  and 
duties  of  which  shall  be  to  acquire  and  to  diffuse  among  the  people 
of  the  United  States  information  on  subjects  connected  with  agricul- 
ture, in  the  most  general  and  comprehensive  sense  of  the  word.' 

i"The  United  States  Department  of  Agriculture,"  Crop  Reporter, 
January,  1901,  pp.  1-2. 

^  The  Government  publication  from  which  the  following  quotations 
are  made  bears  the  title:  Government  Crop  Reports:  Their  Value,  Scope, 
and  Preparation,  and  forms  Circular  17,  of  the  Bureau  of  Statistics,  of 
the  U.  S.  Department  of  Agriculture.  As  the  date  of  the  Circular  is 
September  30,  1908,  some  points  of  detail  may  not  now  be  accurate. 
But  as  our  investigation  will  cover  the  quarter  of  a  century,  1890-1914, 
what  was  said  in  1908  will  give  a  general  idea  of  the  organization  and 
activity  of  the  Department  during  the  period  under  investigation. 


54       Forecasting  the  Yield  and  the  Price  of  Cotton 

The  Commissioner  was  required  by  this  act  to  '  procure  and  preserve 
all  infonnation  concerning  agriculture  which  he  can  obtain  by  means 
of  books,  correspondence,  and  by  practical  and  scientific  experi- 
ments, accurate  records  of  which  experiments  shall  be  kept  in  his 
office,  by  the  collection  of  statistics,  and  by  any  other  appropriate 
means  within  his  power.' 

"The  first  appropriation  for  collecting  agricultural  statistics  by 
the  Department  was  provided  for  by  the  act  of  February  25,  1863, 
which  was  made  in  bullc  for  the  work  of  the  Department,  amounting 
in  all  to  $90,000.  The  then  Commissioner  of  Agriculture  allotted  a 
part  of  this  amount  for  collecting  agricultural  statistics,  and  ap- 
pointed a  statistician  for  that  purpose.  For  the  fiscal  year  ended 
June  30,  1865,  the  first  distinct  and  separate  provision  was  made  for 
collecting  agricultural  statistics  for  information  and  reports,  and 
the  amount  of  $20,000  was  appropriated. 

"From  an  allotment  of  a  few  thousand  dollars  each  year  at  first 
the  crop-reporting  service  has  been  evolved,  perfected,  and  en- 
larged into  the  Bureau  of  Statistics  of  this  Department. 

"The  appropriation  act  for  the  Department  of  Agriculture  for  the 
fiscal  year  ended  June  30,  1908,  carried  appropriations  of  about 
$220,000  for  the  Bureau  of  Statistics,  and  for  the  current  year  the 
appropriation  has  been  increased  to  about  $222,000.  As  the  appro- 
priations for  the  statistical  and  crop-reporting  service  have  been 
gradually  increased  during  the  past  several  years,  the  field  service 
and  organization  of  the  Bureau  have  been  correspondingly  en- 
larged. 

"The  Bureau  of  Statistics  issues  each  month  detailed  reports 
relating  to  agricultural  conditions  throughout  the  United  States,  the 
data  upon  which  they  are  based  being  obtained  through  a  special 
field  service,  a  corps  of  State  statistical  agents,  and  a  large  body  of 
voluntary  correspondents  composed  of  the  following  classes :  County 
correspondents,  township  correspondents,  individual  farmers,  and 
special  cotton  correspondents. 

"The  special  field  service  consists  of  seventeen  traveling  agents, 
each  assigned  to  report  for  a  separate  group  of  States.  These  agents 
are  especially  qualified  by  statistical  training  and  practical  knowl- 
edge of  crops.  They  systematically  travel  over  the  district  assigned 
to  them,  carefully  note  the  development  of  each  crop,  keep  in  touch 
with  best  informed  opinion,  and  render  written  and  telegraphic 
reports  monthly  and  at  such  other  times  as  required. 


The  Government  Crop  Reports  55 

"There  are  forty-five  State  statistical  agents,  each  located  in  a 
different  State.  Each  reports  for  his  State  as  a  whole,  and  main- 
tains a  corps  of  correspondents  entirely  independent  of  those  re- 
porting directly  to  tlae  Department  at  Washington.  These  State 
statistical  correspondents  report  each  month  directly  to  the  State 
agent  on  schedules  furnished  him.  The  reports  are  then  tabulated 
and  weighted  according  to  the  relative  product  or  area  of  the  given 
crop  in  each  county  represented,  and  are  summarized  by  the  State 
agent,  who  coordinates  and  analyses  them  in  the  light  of  his  per- 
sonal knowledge  of  conditions,  and  from  them  prepares  his  reports  to 
the  Department. 

"There  are  approximately  2,800  counties  of  agricultural  im- 
portance in  the  United  States.  In  each  the  Department  has  a 
principal  county  correspondent  who  maintains  an  organization  of 
several  assistants.  These  county  correspondents  are  selected  with 
special  reference  to  their  qualifications  and  constitute  an  efficient 
branch  of  tiie  crop-reporting  service.  They  make  the  county  the 
geographical  unit  of  their  reports,  and,  after  obtaining  data  each 
month  from  theii-  assistants  and  supplementing  these  with  informa- 
tion obtained  from  their  own  observation  and  knowledge,  report 
direct!}^  to  the  Department  at  Washington. 

"In  the  townships  and  voting  precincts  of  the  United  States  in 
which  farming  operations  are  extensively  carried  on  the  Department 
has  township  correspondents  who  make  the  township  or  precinct 
the  geographical  basis  of  reports,  which  they  also  send  directlj^  to 
the  Department  each  month. 

"Finally,  at  the  end  of  the  growing  season  a  large  nmnber  of  in- 
dividual farmers  and  planters  report  on  the  results  of  their  own 
indi^ddual  farming  operations  during  the  year;  valuable  data  are 
also  secured  from  30,000  mills  and  elevators. 

"With  regard  to  cotton,  all  the  information  from  the  foregoing 
sources  is  supplemented  by  that  furnished  by  special  cotton  corre- 
spondents, embracing  a  large  number  of  persons  intimately  con- 
cerned in  the  cotton  industry;  and,  in  addition,  inquiries  in  relation 
to  acreage  and  yield  per  acre  of  cotton  are  addressed  to  the  Bureau 
of  the  Census's  list  of  cotton  ginners  through  the  courtesy  of  that 
Bureau. 

"Eleven  monthly  reports  on  the  principal  crops  are  received 
j^early  from  each  of  the  special  field  agents,  county  correspondents, 
State  statistical  agents,  and  township  correspondents,  and  one  re- 


56        Forecasti7ig  the  Yield  and  the  Price  of  Cotton 

port  relating  to  the  acreage  and  production  of  general  crops  an- 
nually from  individual  farmers. 

"  Six  special  cotton  reports  are  received  during  the  growing  season 
from  the  special  field  agents,  from  the  county  correspondents,  from 
the  State  statistical  agents,  and  from  township  correspondents,  and 
the  first  and  last  of  these  reports  are  supplemented  by  returns  from 
individual  farmers,  special  correspondents,  and  cotton  ginners. 

"In  order  to  prevent  any  possible  access  to  reports  which  relate 
to  speculative  crops,  and  to  render  it  absolutely  impossible  for 
premature  information  to  be  derived  from  them,  all  of  the  reports 
from  the  State  statistical  agents,  as  well  as  those  of  the  special 
field  agents,  are  sent  to  the  Secretary  of  Agriculture  in  specially 
prepared  envelopes  addressed  in  red  ink  with  the  letter  '  A '  plainly 
marked  on  them.  By  an  arrangement  with  the  postal  authorities 
these  envelopes  are  delivered  to  the  Secretary  of  Agriculture  in 
sealed  mail  pouches.  These  pouches  are  opened  only  by  the  Secre- 
tary or  Assistant  Secretary,  and  the  reports,  with  seals  unbroken, 
are  immediately  placed  in  the  safe  in  the  Secretary's  office,  where 
they  remain  sealed  until  the  morning  of  the  day  on  which  the 
Bureau  report  is  issued,  when  they  are  delivered  to  the  Statistician 
by  the  Secretary  or  the  Assistant  Secretary.  The  combination  for 
opening  the  safe  in  which  such  documents  are  kept  is  known  only 
to  the  Secretary  and  the  Assistant  Secretary  of  Agriculture.  Re- 
ports from  special  field  agents  and  State  statistical  agents  residing  at 
points  more  than  500  miles  from  Washington  are  sent  by  telegraph, 
in  cipher.  Those  in  regard  to  speculative  crops  are  addressed  to  the 
Secretary  of  Agriculture. 

"  Reports  from  the  State  statistical  agents  and  special  field  service 
in  relation  to  nonspeculative  crops  are  sent  in  similar  envelopes 
marked  '  B '  to  the  Bureau  of  Statistics  and  are  kept  securely  in  a 
safe  until  the  data  are  required  by  the  Statisticians  in  computing 
estimates  regarding  the  crops  to  which  they  relate.  The  reports 
from  the  county  correspondents,  township  correspondents,  and  other 
voluntary  agents  are  sent  to  the  Chief  of  the  Bureau  of  Statistics  by 
mail  in  sealed  envelopes. 

"The  work  of  making  the  final  crop  estimates  each  month  cul- 
minates at  sessions  of  the  Crop-Reporting  Board,  composed  of  five 
members,  presided  over  by  the  Statistician  and  Chief  of  Bureau  as 
chairman,  whose  services  are  brought  into  requisition  each  crop- 
reporting  day  from   among  the   statisticians  and   officials  of  the 


The  Government  Crop  Reports  57 

Bureau,  and  special  field  and  State  statistical  agents  who  are  called 
to  Washington  for  the  purpose. 

"The  personnel  of  the  Board  is  changed  each  month.  The  meet- 
ings are  held  in  the  office  of  the  Statistician,  which  is  kept  locked 
during  sessions,  no  one  being  allowed  to  enter  or  leave  the  room  or 
the  Bureau,  and  all  telephones  being  disconnected. 

"When  the  Board  has  assembled,  reports  and  telegrams  regarding 
speculative  crops  from  State  and  field  agents,  which  have  been 
placed  miopened  in  a  safe  in  the  office  of  the  Secretary  of  Agricul- 
ture, are  delivered  by  the  Secretary,  opened,  and  tabulated;  and 
the  figures,  by  States,  from  the  several  classes  of  correspondents  and 
agents  relating  to  all  crops 'dealt  with  are  tabulated  in  convenient 
parallel  columns;  the  Board  is  thus  provided  with  several  separate 
estimates  covering  each  State  and  each  separate  crop,  made  in- 
dependently by  the  respective  classes  of  correspondents  and  agents 
of  the  Bureau,  each  reporting  for  a  territor}'-  or  geographical  unit 
with  which  he  is  thoroughly  familiar. 

"With  all  these  data  before  the  Board,  each  individual  member 
computes  independently,  on  a  separate  sheet  or  final  computation 
slip,  his  own  estimate  of  the  acreage,  condition,  or  yield  of  each  crop, 
or  of  the  number,  condition,  etc.,  of  farm  animals  for  each  State 
separately.  These  results  are  then  compared  and  discussed  by  the 
Board  under  the  super^dsion  of  the  chainnan,  and  the  final  figures 
for  each  State  are  decided  upon. 

"The  estimates  by  States  as  finally  determined  by  the  Board  are 
weighted  by  the  acreage  figures  for  the  respective  States,  the  re- 
sult for  the  United  States  being  a  true  weighted  average  for  each 
subject.  Thus,  the  figures  for  the  United  States  are  not  straight 
averages,  which  would  be  secured  by  dividing  the  sum  of  the  State 
averages  by  the  niraiber  of  States;  but  each  State  is  given  its  due 
weight  in  proportion  to  its  productive  area  for  each  crop. 

"Reports  in  relation  to  cotton,  after  being  prepared  by  the  Croi> 
Reporting  Board,  and  personally  approved  by  the  Secretary  of 
Agriculture,  are  issued  on  the  first  or  second  day  of  each  month 
during  the  growing  season,  and  reports  relating  to  the  principal  farm 
crops  and  live  stock  on  the  seventh  or  eiglith  day  of  each  month.  In 
order  that  the  information  contained  in  these  reports  may  be  made 
available  simultaneously  throughout  the  entire  United  States,  thej'' 
are  handed,  at  an  announced  hour  on  report  days,  to  all  applicants 
and  to  the  Western  Union  Telegraph  Company  and  the  Postal 


58       Forecasting  the  Yield  and  the  Price  of  Cotton 

Telegraph  Cable  Company,  who  have  branch  offices  in  the  Depart- 
ment of  Agriculture,  for  transmission  to  the  Exchanges  and  to  the 
press.  These  companies  have  reserved  their  lines  at  the  designated 
time,  and  forward  immediately  the  figures  of  most  interest.  A 
mimeograph  or  multigraph  statement,  also  containing  such  esti- 
mates of  condition  or  actual  production,  together  with  the  corre- 
sponding estimates  of  former  years  for  comparative  purposes,  is 
prepared  and  sent  immediately  to  Exchanges,  newspaper  publica- 
tions, and  individuals.  The  same  day  printed  cards  containing  the 
essential  facts  concerning  the  most  important  crops  of  the  report  are 
mailed  to  the  77,000  post-offices  throughout  the  United  States  for 
public  display,  thus  placing  most  valuable  information  within  the 
farmer's  immediate  reach. 

"Promj)tly  after  the  issuing  of  the  report,  it,  together  with  other 
statistical  information  of  value  to  the  farmer  and  the  country  at 
large,  is  published  in  the  Crop  Reporter,  an  eight-page  publication  of 
the  Bureau  of  Statistics,  under  the  authority  of  the  Secretary  of 
Agriculture.  An  edition  of  over  120,000  copies  is  distributed  to  the 
correspondents  and  other  interested  parties  throughout  the  United 
States  each  month." 

Technical   Terms:  Norinal,   Condition,   Indicated   Yield 

per  Acre 

To  understand  the  official  method  of  forecasting  the 
size  of  agricultural  crops,  one  must  have  clearly  in  mind 
the  technical  meaning  of  the  terms  normal,  condition, 
and  indicated  yield  per  acre.  The  correspondents  of  the 
Bureau  of  Statistics  of  the  Department  of  Agriculture 
are  instructed  to  assume  that  a  normal  crop  is  to  be 
represented  by  100,  and  they  are  asked  to  express  the 
condition  of  the  crop  in  their  respective  districts,  during 
successive  months,  as  percentages  of  the  normal.  From 
these  figures  of  condition  supplied  by  its  correspondents 
and  agents,  the  Bureau  of  Statistics  computes  the 
indicated  yield  per  acre  for  the  several  states,  and  for 
the  whole  country. 


The  Government  Crop  Reports  59 

But  what  is  a  normal  crop?  Although  the  degree  of 
efficiency  of  the  crop-reporting  service  is  largely  de- 
pendent upon  an  accurate  definition  and  sufficient 
understanding  of  this  fundamental  term,  the  Bureau 
of  Statistics  has  been  very  slow  to  give  an  adequate  de- 
scription of  its  meaning.  The  official  instruction  which 
for  a  long  time  was  given  to  the  correspondents  of  the 
Department  is  here  quoted  at  length: 

The  Normal.  "So  many  of  the  reports  of  the  Statistician  of  the 
Department  of  Agriculture  are  based  upon  a  comparison  with  the 
'normal'  that  it  is  a  matter  of  the  greatest  importance  that  there 
should  be  a  clear  understanding  of  what  the  normal  really  means. 

"To  begin  with,  a  normal  condition  is  not  an  average  condition, 
but  a  condition  above  the  average,  giving  promise  of  more  than  an 
average  crop. 

"Furthermore,  a  normal  condition  does  not  indicate  a  perfect 
crop,  or  a  crop  that  is  or  promises  to  be  the  very  largest  in  quantity 
and  the  very  best  in  quality  that  the  region  reported  upon  may  be 
considered  capable  of  producing.  The  normal  indicates  something 
less  than  this,  and  thus  comes  between  the  average  and  the  possible 
maximum,  being  greater  than  the  fonner  and  less  than  the  latter. 

"The  nomial  may  be  described  as  a  condition  of  perfect  health- 
fulness,  unmipaired  by  drought,  hail,  insects,  or  other  injurious 
agency,  and  with  such  growth  and  de\-clopment  as  maj^  reasonabh^ 
be  looked  for  under  these  favorable  conditions.  As  stated  in  the 
instruction  to  correspondents,  it  does  not  represent  a  crop  of  extraor- 
dinary character,  such  as  may  be  produced  here  and  there  by  the 
special  effort  of  some  highly  skilled  farmer  with  abundant  means,  or 
such  as  may  be  grown  on  a  bit  of  land  of  extraordinary  fertility,  or 
even  such  as  may  be  grown  quite  extensi^^ely  once  in  a  dozen  years  in 
a  season  that  is  extraordinarily  fa\'orable  to  the  croj)  to  be  raised. 
A  normal  crop,  in  short,  is  neither  dehcient  on  the  one  hand  nor 
extraordinarily  heavj^  on  the  other.  While  a  normal  condition  is  l)ut 
rarely  reported  for  the  entire  corn,  wheat,  cotton,  or  other  crop 
area,  at  the  same  time  or  in  the  same  year,  its  local  occurrence  is  by 
no  means  uncommon,  and  whenever  it  is  found  to  exist,  it  sliould  be 
indicated  by  the  number  100. 

"Sometimes  a  favorable  season   for  j)laiitiiig  is  followed   l)y  a 


60       Forecasting  the  Yield  and  the  Price  of  Cotton 

favorable  growing  season,  with  no  blight  and  no  depredations  by- 
insects,  the  result  being  a  normal  condition.  At  other  times  the 
normal  may  be  maintained  by  conditions  that  are  exceptionally 
favorable  in  one  or  more  particulars  counterbalancing  conditions 
that  are  unfavorable  in  other  particulars.  Thus,  a  crop  may  have 
had  such  an  unusually  good  start  that  it  may  pass  without  injury 
through  a  period  of  drought  that  would  otherwise  have  proved 
disastrous  to  it,  or  its  more  than  ordinary  vigor  and  potentiality  may 
fully  offset  some  slight  injury  from  insects. 

"  The  noiTual  not  being  everywhere  the  same,  in  determining  how 
near  the  condition  of  any  given  crop  is  to  the  normal,  correspondents 
will  usually  find  it  an  advantage  to  have  a  definite  idea  of  what  yield 
per  acre  would  constitute  a  full  noi-mal  crop  in  their  respective  dis- 
tricts; that  is,  how  many  bushels,  pounds,  or  tons  per  acre  of  a 
particular  crop  would  be  produced  in  a  season  that  was  distinctly 
but  not  exceptionally  favorable.  In  a  region  where  30  bushels  of 
corn,  may  be  taken  as  the  nonnal,  a  condition  of  90  would  give  a 
prospect  of  a  crop  of  27  bushels,  and  80  a  crop  of  24  bushels.  If  40 
bushels  be  considered  the  normal  yield,  90  (or  ten  per  cent  less  than 
the  normal)  would  indicate  a  crop  of  36  bushels,  80  one  of  32  bushels, 
70  one  of  28  bushels. 

"For  the  reason  that  the  nonnal,  represented  by  100,  does  not 
indicate  a  perfect  or  the  largest  possible  crop,  it  may  occasionally 
be  exceeded.  The  condition  may  be  so  exceptionally  favorable  as 
to  promise  a  crop  that  will  exceed  the  normal,  and  it  will  accordingly 
have  to  be  expressed  by  105,  110,  or  whatever  other  figures  may  seem 
warranted  by  the  facts;  105  representing  five  per  cent  above  the 
normal,  110  ten  per  cent,  and  so  forth."  ^ 

The  least  that  can  be  said  about  this  definition  of 
normal  is  that,  as  the  individual  farmer-correspond- 
ents must  express  the  current  condition  of  the  crop  as  a 
percentage  of  the  normal,  the  official  Bureau  leaves 
much  to  the  individual  farmers  to  determine.  As 
late  as  August,  1916,  a  writer  of  great  influence  in  the 
cotton  trade  has  condemned  the  whole  crop-reporting 
service  because  of  the  lack  of  precision  in  the  instruc- 

1  Crop  Reporter,  May,  1899,  p.  3.  Cf.  Circular  17  of  the  Bureau  of 
Statistics,  pp.  12-13. 


The  Government  Crop  Reports  61 

tions  that  are  sent  to  those  who  supply  the  primary 
statistical  data: 

"Those  who  report  for  the  Crop  Estimating  Board 
are  asked  to  make  a  mental  comparison  between  exist- 
ing conditions  and  an  imaginary  normal.  They  are 
instructed  to  assume  that  this  indefinable  normal  is 
represented  by  100  and  to  describe  the  present  and  its 
promise  in  figures  that  are  supposed  to  be  a  percentage 
of  an  impossible  perfection.  The  very  difficulty  of 
stating  the  theory  upon  which  the  reports  are  compiled 
shows  how  misleading  they  may  be,  but  the  practical 
impossibility  of  applying  the  theory  utterly  shatters 
any  claims  that  such  findings  are  entitled  to  scientific 
consideration."  ^ 

Although,  as  we  ha^"e  seen,  the  reports  on  the  con- 
dition of  the  growing  crops  have  been  issued  contin- 
uously since  1866,  the  Government  authorities,  until 
1911,  systematically  refused  to  say  what  was  to  be 
inferred  from  their  laboriously  compiled  tables.  We 
have  the  repeated  statement  that  "the  Department, 
as  is  well  known,  makes  no  attempt  to  estimate  in  ad- 
vance the  probable  yield  of  any  agricultural  product."  ^ 
"The  Department's  reports  previous  to  harvest" 
are  intended  "simply  as  a  general  epitome  of  the  crop 
situation."  '  But  the  Department  has  been  aware  all 
along  that  the  reports  "are  interpreted  as  furnishing 
a  basis  for  quantitative  forecasts  of  yield."  -     It  is 

'  Theodore  H.  Price:  "The  Value  and  Defects  of  Governnient  Crop 
Reports,"  Commerce  and  Finance,  August  16,  19i6,  p.  915.  Mr. 
Price's  strictures  with  reference  to  the  "indefinable  noi-rnal"  do  not 
hold  with  the  same  degree  of  force  since  the  imjirovemcmt  in  the  crop- 
reporting  service  in  1911.  Whether  the  reports  have  any  scientific 
value  will  be  revealed  in  the  course  of  this  chapt(M'. 

2  Crop  Reporter,  May,  1900,  p.  6,  and  May,  1902,  p.  4. 


62        Forecasting  the  Yield  and  the  Price  of  Cotton 

surely  not  amiss  to  say  that  the  appropriations  of  pubhc 
funds  for  the  crop-reporting  service  have  been  made 
not  because  the  Department  has  suppUed  "simply 
a  general  epitome  of  the  crop  situation,"  but  because 
the  claim  has  been  urged  that  the  crop-reporting  serv- 
ice gives  the  public,  and  particularly  the  farmers, 
''early  information  concerning  the  supply"  ^  of  agri- 
cultural products. 

Until  the  Department  of  Agriculture  told  us  what 
its  crop  reports  meant  and  how  its  elaborate  tables 
as  to  crop  condition  were  to  be  utilized  to  forecast 
the  probable  yield  per  acre,  it  was  not  possible  to  test 
the  efficiency  and  utiUty  of  the  crop-reporting  service. 
The  Department,  since  1911,  has  given  us  the  needed 
information,  and  before  we  pass  to  the  actual  testing 
of  the  degree  of  accuracy  in  the  official  forecasts  we 
shall  quote  at  length  the  official  description  of  the  "In- 
terpretation of  Crop  Condition  Figures." 

"  The  Bureau  of  Statistics  has  this  year  for  the  first  time  given  a 
quantitative  interpretation  to  its  monthly  figures  relating  to  the 
condition  of  growing  crops;  that  is,  has  indicated  the  jdeld  which 
the  condition  figures  suggest.  Much  interest  is  manifested  as  to 
the  method  used  in  making  such  interpretations. 

"It  is  assumed,  in  the  first  place,  that  average  conditions  at  any 
time  are  indicative  of  average  yields  per  acre;  that  conditions  above 
an  average  at  any  time  are  indicative  of  yields  above  the  average; 
and  conditions  below  the  average  at  any  time  are  indicative  of 
yields  below  the  average.  If  at  any  time  the  condition  of  a  growing 
crop  is  5  per  cent  above  the  average  condition  for  such  tune,  it  is 
assumed  that  the  yield  is  more  likely  to  be  5  per  cent  above  the 
average  yield  than  any  other  amount.  If  the  condition  at  any  time 
is  10  per  cent  below  the  average  for  such  time,  it  is  assumed  that 

1  "Government  Crop  Reports:  Their  Value,  Scope,  and  Preparation." 
U.  S.  Department  of  Agriculture.  Bureau  of  Statistics.  Circular  17, 
p.  7. 


The  Government  Crop  Reports  63 

the  yield  is  more  likely  to  be  10  per  cent  below  the  average  than 
any  other  amount. 

"As  a  growing  crop  progresses  toward  maturity,  its  relation  to  an 
average  condition  is  almost  constantly  changing;  if  the  growing 
period  becomes  more  favorable  than  the  average,  the  prosj^ects 
improve  and  the  indicated  yield  enlarges;  as  the  growing  period  be- 
comes less  favorable  than  the  average,  the  prospect  diminishes  and 
the  indicated  yield  lessens. 

"In  interpreting  the  condition  figures  it  is  necessary  to  deter- 
mine what  is  an  average  condition  at  any  time  and  what  is  the 
corresponding  average  yield.  Different  results  will  be  obtained  by 
using  different  bases.  For  instance,  the  condition  of  spring  wheat 
on  July  1  in  the  last  5  years  (84.5  per  cent  of  nonnal)  averaged 
nearly  4  per  cent  lower  than  the  average  for  the  last  10  years  (87.8) ; 
and  the  average  yield  per  acre  in  the  last  5  years  (13.5  bushels)  is 
about  2  per  cent  lower  than  the  average  for  the  last  10  j'-ears  (13.8 
bushels) . 

"The  objection  to  a  10-year  basis  for  determining  either  the 
average  or  nonnal  yield  of  crops  is  that  there  is  a  gradual  tendency 
of  the  average  or  normal  yield  per  acre  for  the  United  States  to  in- 
crease from  year  to  year,  and  therefore  an  average  based  upon  a 
long  series  of  years  will  be  too  low.  For  instance,  the  calculated 
equivalent  of  100  condition  for  winter  wheat  at  harvest  tune  in  the 
last  3  years  averaged  about  18.8  bushels,  the  average  of  the  last 
6  years  was  about  18.6  bushels,  and  for  the  last  10  years.  17.9 
bushels. 

"On  the  other  hand,  a  5-year  basis  includes  so  few  years  that  one 
extreme  or  abnormal  year  in  the  5  may  so  affect  the  average  as  to 
make  it  not  representative  of  general  average  condition.  For  in- 
stance, the  jneld  per  acre  of  flaxseed  in  1910  was  abnonnally  low, 
4.8  bushels,  as  compared  with  9.4  in  1909,  9.6  in  1908,  9  in  1907,  and 
10.2  in  1906,  the  average  of  the  5  years  being  8.6,  which  is  lower 
than  any  year  included  in  the  average  except  1910;  the  year  1910, 
therefore,  ought  to  be  omitted  in  obtaining  a  figure  representing 
average  conditions. 

"After  a  study  of  the  results  obtaiiu^d  fi-om  using  5  j^ears  and  10 
years,  respectively,  for  basing  an  average,  the  advantage  is  found 
to  be  slightly  in  favor  of  the  5-year  basis.  In  using  the  5-year  basis, 
however,  it  is  proper  to  omit  years  of  abnormal  conditions. 

"The  process  in  the  interpretation  may  1)0  explained  I)y  an  ex- 


64       Forecasting  the  Yield  and  the  Price  of  Cotton 

ample.  The  condition  of  com  on  July  1,  1911,  was  80.1  per  cent  of  a 
normal  condition;  in  the  last  5  years  the  condition  has  averaged 
85  per  cent  of  a  nonnal  condition;  thus  the  condition  on  July  1  is 
5.8  per  cent  below  the  average  condition  (80.1  being  94.2  per  cent 
of  85),  and  suggests  a  yield  of  5.8  per  cent  below  the  average.  In 
the  last  5  years  the  yield  averaged  about  27.1  bushels;  94.2  per  cent 
of  27.1  bushels  (94.2  X  27.1)  is  nearly  25.5  bushels;  therefore  condi- 
tions are  said  to  indicate  a  yield  of  25.5  bushels.  That  is;  if  the  con- 
dition of  the  corn  crop  be  5.8  per  cent  below  the  average  at  harvest 
time,  a  3deld  of  25.5  bushels  is  the  most  reasonable  expectation;  if 
less  than  the  average  adversity  befall  the  crop  before  harvest,  a 
larger  yield  may  be  expected;  if  more  than  the  average  adversity 
befall  the  crop,  a  j'ield  less  than  25.5  bushels  may  be  expected. 

"Another  method  of  interpretation  of  the  bureau's  condition  re- 
port has  been  used  by  some  private  statisticians,  wliich  may  be  ex- 
plained here  briefly  by  an  example.  The  condition  of  the  corn  crop 
on  July  1,  1911,  is  80.1  per  cent  of  normal;  the  average  condition 
of  the  corn  crop  on  October  1  for  the  last  5  years  has  been  80  per 
cent  of  a  normal;  thus  the  condition  on  July  1  (80.1)  is  0.1  per  cent 
above  the  average  condition  (80)  on  October  1  (the  October  report 
being  the  nearest  to  the  harvest  condition).  The  average  yield 
being  27.1  bushels,  0.1  per  cent  above  average  would  be  nearly 
27.1  bushels,  the  yield  indicated  on  this  basis,  as  against  25.5 
bushels  indicated  by  the  method  adopted  by  the  Bureau  of  Statistics. 

"The  difference  between  the  two  methods  is  this:  By  the  one 
adopted  by  the  bureau  it  is  assumed  that  from  July  1  to  harvest  the 
average  amoimt  of  variation  will  occur.  (The  5-year  average  condi- 
tion on  July  1  is  85  per  cent  of  a  normal;  the  5-year  average  condi- 
tion on  October  1  is  80.)  By  the  other  method  it  is  assumed  that  no 
variation  in  condition  will  occur,  notwithstanding  that  in  the  last 
5  years  the  average  change  has  been  from  85  to  80,  or  a  decline. 

"  The  difference  between  the  two  methods  of  interpretation  may 
also  be  shown  as  follows,  using  the  same  example:  The  average  con- 
dition of  corn  July  1  is  85  and  the  average  yield  27.1,  hence  the 
equivalent  of  100  on  July  1  is  31.9  bushels  (27.1  X  100  ^  85);  hence 
a  condition  of  80.1  on  July  1  indicates  a  yield  of  25.5  bushels  (31.9  X 
80.1  -^  100).  This  is  the  method  adopted  by  the  Bureau  of 
Statistics. 

"The  other  method  referred  to,  used  by  some  statisticians,  is  as 
follows:  The  average  condition  of  corn  on  October  1  (the  condition 


The  Government  Crop  Reports  65 

report  nearest  to  time  of  harvest)  is  80  per  cent,  average  yield, 
27.1  bushels;  hence  the  equivalent  of  100  on  October  1  (27.1  X 
100  -i-  80)  is  33.9  bushels;  this  equivalent  of  100  is  used  throughout 
the  growing  season  as  the  equivalent  of  100,  and  hence  on  July  1 
when  the  condition  is  80.1  it  is  interpreted  as  indicating  a  yield  of 
27.1  bushels  (33.9  X  80.1  -  100). 

"The  difference  between  the  two  methods  is  that  the  one  adopted 
by  the  bureau  allows  for  the  natural  variation  as  the  season  pro- 
gresses, while  the  other  does  not. 

"It  may  be  pertinent  to  observe,  considering  the  interpretation  of 
crop  condition  figures,  that  the  higher  the  condition  of  a  crop  the 
more  sensitive  it  is;  that  is,  liable  to  a  decline  before  harvest.  For 
example,  of  the  last  10  years,  the  5  which  give  the  highest  condition 
of  winter  wheat  on  May  1  averaged  91.8  per  cent  of  normal,  and  the 
remainmg  5  years  of  lowest  condition  on  May  1  averaged  80.3.  The 
5  years  which  averaged  91.8  per  cent  on  May  1  averaged  83.2  per 
.  cent  on  July  1,  a  drop  of  8.6  points,  or  9.4  per  cent;  the  5  years  which 
averaged  80.3  per  cent  on  May  1  averaged  79.6  on  July  1,  a  drop  of 
only  0.7  point,  or  0.9  per  cent.  Neither  method  described  takes 
into  account  this  factor."  '■ 


The  Accuracy  of  Forecasts  Tested 

If,  for  a  moment,  we  review  the  description  given  by 
the  Bureau  of  Statistics  of  its  method  of  forecasting 
the  probable  yield  per  acre  of  a  crop,  we  shall  see  that 
no  reason  is  given  for  preferring  the  five  years  average 
over  the  ten  years  average  as  the  basis  of  prediction, 
except  that  the  former  gives  a  ''slightly  better  result." 
In  what  way  the  result  of  the  five  years  method  is 
better  than  the  result  of  the  ten  years  method  is  not 
stated,  nor  is  there  indicated  any  way  by  which  we  may 
determine  how  much  better  one  method  is  than  another. 
But  it  must  be  very  clear  that,  for  business  and  for 
scientific  purposes,  the  measurement  of  the  degree  of 

'  Crop  Reporter,  July,  1911,  pp.  .53-55. 


66       Forecasting  the  Yield  and  the  Price  of  Cotton 

accuracy  and  reliability  of  forecasts  is  of  the  first  im- 
portance. 

Without  a  measure  of  the  degree  of  accuracy  and  re- 
liabiUty  of  the  forecasts  the  Crop-Reporting  Board  has 
no  proper  ground  of  choice  between  different  methods 
of  forecasting;  farmers,  brokers,  and  consumers  are 
without  adequate  guidance  in  the  planning  of  their 
enterprises ;  and  scientific  economists  have  no  empirical 
tests  of  the  degree  of  error  in  their  analyses  of  the  in- 
terrelation of  phenomena.  The  Crop-Reporting  Board, 
as  we  have  seen,  uses  a  five  years  average  in  preference 
to  one  of  ten  years.  But  would  not  a  four  years  aver- 
age, or  a  three  years  average,  give  better  results  than 
the  five  years  average?  The  Bureau  of  Statistics  issues 
annually  six  cotton  reports,  beginning  with  May.  WTiat 
is  the  value  of  these  several  reports  as  forecasts?  Do 
the  predictions  become  increasingly  accurate  with  the 
approach  of  harvest?  Are  the  reports  for  all  of  the 
months  worthy  of  confidence,  or  are  some  of  them  so 
misleading  as  to  suggest  the  wisdom  of  their  discontin- 
uance? If  all  of  the  forecasts  are  affected  with  error, 
is  there  a  tendency  of  the  error  to  favor  the  manu- 
facturer or  the  farmer,  and  what  are  the  risks  assumed 
in  planning  one's  enterprise  according  to  the  forecasts? 
The  answers  to  all  of  these  questions  become  possible 
when  we  have  found  an  adequate  method  of  measuring 
the  degree  of  accuracy  in  the  forecasts.  To  this  prob- 
lem we  now  turn. 

The  method  of  the  Crop-Reporting  Board  in  making 
the  forecast  of  the  yield  per  acre  of  cotton  is,  as  we 
have  seen,  to  assume  that  the  ratio  of  the  yield  per 
acre  for  any  given  year  to  the  mean  yield  per  acre  for 


The  Government  Crop  Reports  67 

the  preceding  five  years  is  equal  to  the  ratio  of  the  con- 
dition of  any  given  month  to  the  mean  condition  for  the 
gi\'en  month  during  the  preceding  five  years.  The 
method  may  be  put  in  the  form  of  symbols :  Let  Y  =  the 
yield  per  acre  of  cotton  for  the  current  year;  Y^  =  the 
mean  yield  per  acre  of  cotton  for  the  preceding  five 
years;  C  =  the  condition  of  the  crop  for  the  current 
month;  C^  the  mean  condition  for  the  same  month 
during  the  preceding  five  years.  Then,  according  to 
the  assumption  of  the  Crop-Reporting  Board,  ^/ Fs 
=  ^ICh-  We  shall  call  ^7  F5  the  yield-ratio ;  and  C'/c's, 
the  condition-ratio. 

In  Table  5  we  have  the  record,  for  a  quarter  of 
a  century,  of  the  actual  yield  per  acre  of  cotton, 
and  the  yield  as  predicted,  according  to  the  method 
of  the  Crop-Reporting  Board,  from  the  condition 
of  the  crop  on  the  25th  of  September.  The  column 
marked  ^/Cb  gives  the  ratio  of  the  condition  of  the  crop, 
at  the  end  of  September  of  any  given  year,  to  the  mean 
condition  of  the  crop,  at  the  end  of  September,  during 
the  preceding  five  years.  According  to  the  method  of 
the  Crop-Reporting  Board,  the  ratio  C*/^,_  is  the  prob- 
able ratio  that  the  yield  per  acre  of  any  given  year 
should  bear  to  the  mean  yield  of  the  preceding  five 
years.  But  the  column  marked  ^/y.^  gives  the  ratio 
of  the  actual  yield  per  acre  to  the  mean  yield  per  acre 
of  the  preceding  five  years.  An  inspection  of  the  Table 
shows  that  the  two  series  —  the  predicted  series  and  the 
actual  series  —  are  not  equal.  Is  there  any  relation 
between  the  two  series,  and  if  so,  how  closely  are  they 
related,  and  what  is  the  error  made  in  using  the  condi- 
tion-series to  forecast  the  yield-series? 


68      Forecasting  the  Yield  and  the  Price  of  Cotton 


TABLE  5.  —  The  Actual  Yield  Per  Acre  of  Cotton  in  the 
United  States  Compared  with  the  Yield  as  Forecast  from 
THE  September  Condition  of  the  Crop 


Year 

Condition 

of  crop 

September  25 

C 

VTean  condition 
September  25, 
for  the  preced- 
ing 5  years 

'U 

Yield   per  acre 

of  cotton 

in  pounds  of 

lint 

F 

Mean  yield 

per  acre  of 

cotton,  in 

pounds  of 

lint,  for  the 

preceding 

5  years 

'fa 

V-y. 

1885 
6 

7 

78.0 

163.9 

79.3 

169.5 

76.5 

182.8 

8 

78.9 

180.4 

9 

81.5 

177.0 

1890 

80.0 

78.8 

101.5 

187.0 

174.7 

106.8 

1 

75.7 

79.2 

95.6 

179.4 

179.3 

100.1 

2 

73.3 

78.5 

93.4 

209.2 

181.3 

115.4 

3 

70.7 

77.9 

90.8 

148.8 

186.6 

79.7 

4 

82.7 

76.2 

108.5 

191.7 

180.3 

106.3 

5 

65.1 

76.5 

85.1 

155.6 

183.2 

84.9 

6 

60.7 

73 . 5 

82.6 

124.1 

176.9 

70.2 

7 

70.0 

70.5 

99.3 

181.9 

165.9 

109.6 

8 

75.4 

69.8 

108.0 

219.0 

160.4 

136.5 

9 

62.4 

70.8 

88.1 

184.1 

174.5 

105.5 

1900 

67.0 

66.7 

100.4 

194.4 

172.9 

112.4 

1 

61.4 

67.1 

91.5 

169.0 

180.7 

93.5 

2 

58.3 

67.2 

86.8 

188.5 

189.7 

99.4 

3 

65.1 

64.9 

100.3 

174.5 

191.0 

91.4 

4 

75.8 

62.8 

120.7 

204.9 

182.1 

112.5 

5 

71.2 

65.5 

108.7 

186.1 

186.3 

99.9 

6 

71.6 

66.4 

107.8 

202.5 

184.6 

109.7 

7 

67.7 

68.4 

99.0 

178.3 

191.3 

93.2 

8 

69.7 

70.3 

99.1 

194.9 

189.3 

103.0 

9 

58.5 

71.2 

82.2 

154.3 

193.3 

79.8 

1910 

65.9 

67.7 

97.3 

170.7 

183.2 

93.2 

11 

71.1 

66.7 

106.6 

207.7 

180.1 

115.3 

12 

69.6 

66.6 

104.5 

190.9 

181.2 

105.4 

13 

64.1 

67.0 

95.7 

182.0 

183.7 

99.1 

14 

73.5 

65.8 

111.7 

207.9 

181.1 

114.8 

The  Government  Crop  Reports  69 

By  utilizing  the  methods  described  in  the  preceding 
chapter  we  not  only  may  answer  these  questions,  but 
we  shall  gain  additional  information  of  very  great 
importance.  Suppose  we  let  the  values  in  the  ^ Iy^ 
series  be  represented  by  y  and  the  values  in  the  ^ICh 
series  be  represented  by  x.  Then,  we  know  from  the 
theory  of  correlation,  if  the  association  between  the 
variables  is  linear,  the  closeness  of  their  relation  is 
measured  by  the  coefficient  of  correlation  r,  and  the 
straight  line  connecting  the  values  of  y  with  the  values 

of  X  is  described  by  the  equation  {y  —  y)  =  r  —  (x  —  x), 

where  y,  x  and  o-y,  cr^  are  the  means  and  the  stand- 
ard deviations,  respectively,  of  the  y's  and  x's.  By 
putting  our  reasoning  into  symbolic  form  we  see  the 
implied  assumptions  in  the  method  of  the  Crop-Re- 
porting Board.    The  method  assumes  that  y  =  x,  and, 

consequently,  it  implicitly  assumes  (1)  that  r  —  =  1, 

o-.r 

and  (2)  that  y  ^  x.    The  outcome  of  these  assumptions 

we  shall  presently  consider. 

As  the  number  of  observations  in  this  case  is  small, 
extending  only  through  twenty-five  years,  the  method 
of  computing  the  coefficient  of  correlation  differs 
slightly  from  the  method  that  was  illustrated  in  Chap- 
ter II;  and  for  this  reason,  the  process  of  computation 
is  completely  exemplified  in  Table  6.  We  find  that  the 
relation  between  the  two  variables  y  and  x,  for  the 
month  of  September,  is,  r  =  .085. 

The  graph  in  Figure  8  shows  the  scatter  diagram 
connecting  the  yield-series  with  the  condition-series. 
The  equation  to  the  straight  line  connecting  the  two 


70       Forecasting  the  Yield  arid  the  Price  of  Cotton 


TABLE  6.  —  CoRRKLATioN    OF    THE    Septemher    Condition-Ratio 
AND  THE  Yield-Ratio  of  Cotton 


Year 

September 

I'ondition- 

Ratio 

Yield- 
Ratio 

Arbitrary 
Origin  of 
Condition- 
Ratio  at 
100 
x' 

Arbitrary- 
Origin  of 

Yield- 
Ratio  at 
100 

u' 

(x')2 

(u'y- 

x'y' 

+  x'y' 

—  x'y' 

1890 

1 

101.5 

106.8 

1.5 

6.8 

2.25 

46.24 

10.20 

95.6 

100.1 

—  4.4 

0.1 

19.36 

.01 

.44 
101.64 

2 
3 

93  4 

115.4 

—  6.6 

15.4 

43 .  56 

237.16 
412.09 

90.8 

79 .  7 

—  9  2 

—  20.3 

84 .  64 

186.70 

4 
5 

108.5 

106.3 

8.5 

6.3 

72.25 

39 .  69 

53 .  55 

85 . 1 

S4 . 9 

—  14.9 

—  15.1 

222.01 

228.01 

224  99 

518,52 

6 

82.6 
99  3 

70.2 

—  17.4 

—  29.8 

302 . 76 

888 . 04 

7 

109.6 

—  0.7 

9.6 

.49 

92.16 

6.72 
65.45 

8 

108.0 
88.1 

136.5 

8.0 

36.5 

64 .  00 
141.61 

1332.25 

292  00 

9 
1900 

105.5 

—  11  9 

5.5 

30 .  25 

100.4 

112  4 

0.4 

12.4 

.16 

153   76 

4.96 

1 
2 
3 
4 
5 

91.5 

93 . 5 

—  8.5 

—  6.5 

72.25 

42.25 

55.25 

86.8 

99.4 

—  13.2 

—  0.6 

174.24 

.36 

7.92 

100.3 

91.4 

0.3 

—  8.6 

.09 

73.96 

2,58 

.87 

120.7 

112.5 

20.7 

12.5 

428 . 49 

156.25 

258.75 

108.7 

99.9 

8.7 

—  0  1 

75 .  69 

60 .  84 

.01 

« 

107.8 

109.7 

7.8 

9.7 

94 .  09 

75 .  66 

7 
8 

99  0 

93.2 

—  1.0 

—  6.8 

1 .  00 

46  24 

6.80 

99.1 

103.0 

—  0  9 

3.0 

.81 

9 .  00 

2.70 

9 

82.2 

79  8 

—  17.8 

—  20.2 

316.84 
7.29 

408  04 

359 . 56 

1910 
11 
12 

97  3 

93  2 

—  2.7 

—  6.8 

46  24 

18  36 

106.6 

115  3 

6 . 6 

15  3 
5.4 

•43 .  56 
20 .  25 

234  09 

100.9S 

104   5 

105  4 

4.5 

29.16 

24 .  30 

13 

95.7 

99.1 

—  4.3 

11.7 

—  0.9 

18.49 

.81 

3.87 

14 

111.7 

114  8 

14.8 

136.89 

219.04 

173.16 

Totals 

78.7 
—  113  5 

153.3 
—  115.7 

2309.72 

4819.20 

2375 . 59 

180.40 

—  34 . 8 

37.6 

Using  the  same  symbols  that  we  employed  in  Chapter  II,  we  have 

—  34.8                          2(.r')-        2309.72 
d^  = =  —1.392;—-^  =  -—  =  92.3928; 


o-.r  =  \ -^ d:r  =  9.51; 


25 


37.6 

dy  =  =  1.504; 

^         25  '     25 


2(.r'/y')  _  2375  .  59  —  180 .  40 

25       ~  25 

2(2/')'       4819.20 


=  87.8076; 


=  192.7680; 


o-^  =  Y 


/^(y'y-        ,..,        _  __  2(.r/,)        2(x'y') 


25 


dy  =  13.80 


dsdu  =  89.9012; 


25a-j.o-,^ 


The  Government  Crop  Reports 


71 


/3S 

-T 

1                      1 r 

■■  ■  1 

IZ5 

- 

/■ 

IIS 

_ 

• 

^,/^. 

^ 

"          /X 

-o 

'    '/f        ' 

'  \y^    * 

- 

]  /    .. 

- 

1 

/ 

J 

• 

es 

■   7 

■ 

75 

1 

1       1        1 

.    .               1 

BO  90  100  I/O  I20 

Vo/ue  of  the  series  /^ 

Figure  8.  —  Correlation 'of  the  actual  yield-ratios  of  cotton  with  the 
forecasts  from  the  September  condition  of  the  crop. 

Equation  to  the  straight  line,  y  =  .994x  +  3.49,  where  y  =  the  probable 
value  of  ^/y5  and  x  =  ^/Co- 


72       Forecasting  the  Yield  and  the  Price  of  Cotton 
series,  which  is  given  on  Figure  8,  was  computed  from 
(y  —  y)  =  '^~  (^  ""  ^)-     For  any  given  value  of  x,  we 

may  find  the  corresponding  most  probable  value  of 
y,  either  by  substituting  for  x  in  the  equation  to  the 
straight  line  and  then  solving  for  y,  or,  by  using  the 
graph  of  Figure  8  obtain  the  ordinate  of  the  straight 
line  corresponding  to  the  given  abscissa.  Further- 
more, we  are  able  to  measure  the  scatter  of  the  ob- 
servations about  the  straight  line  from  the  formula, 
S  =  (Ty'^l  —  r-.  In  the  particular  case  before  us, 
ay  =  13.80;  r  =  .685;  and,  consequently, /S  =  10.06.  In 
brief,  we  know  the  closeness  of  the  relation  between  the 
two  series  from  the  value  of  r  =  .685;  we  know  the  law 
connecting  the  two  series  from  the  equation,  y  =  .99  x 
-h  3.49;  and  we  know  the  magnitude  of  the  error  made 
in  using  this  law  as  a  formula  to  predict  the  probable 
yield  per  acre,  because  S  =  o-^^l  — r-  =  10.06. 

If  we  examine  the  degree  of  accuracy  of  the  forecast 
obtained  by  using  the  method  of  the  Bureau  of  Statis- 
tics, we  shall  find  that,  for  this  month  of  September, 
the  results  agree  very  closely  with  the  value  of  S  which 
we  have  obtained  by  the  method  of  correlation.  The 
forecast-series  in  Table  5  is  given  in  the  column  marked 
^/Cb  ^iid  the  actual  series  is  given  in  the  column  marked 
"^/Yb-  If  we  take  the  sum  of  the  squares  of  the  differ- 
ences between  ^^/y",-,  and  CICb>  ^^^  divide  by  the  number 
of  cases,  we  shall  have  the  mean  square  of  the  deviations 
of  the  actual  series  from  the  theoretical  series,  and  this 
value  is  comparable  with  S^.  If  we  put  S'  equal  to  the 
square-root  of  the  mean  square  of  the  deviations  of  the 
actual  series  from  the  predicted  series,  we  may  write, 


The  Government  Crop  Reports  73 


iS'  =  y  -j  -^^ — ^^^ ^^^  y ,  where  N  is  the  num- 
ber of  years  through  which  the  series  extends,  and 
2(^/^5  —  ^Ic^-  indicates  the  sum  of  the  squares  of 
the  deviations,  during  the  several  years,  of  the  actual 
series  from  the  predicted  series.  The  value  of  S'  for 
the  month  of  September  is,  S'  =  10.47;  and  the  value 
of  >S,  we  found  just  now,  was  S>  =  10.06.  We  see,  accord- 
ingly, that  for  this  month  of  September,  the  accuracy 
of  the  forecast  by  the  method  of  the  Bureau  of  Statis- 
tics is  about  as  great  as  the  accuracy  of  the  method  of 
correlation.    A  moment  ago  we  found  that  the  implicit 

assumptions  in  the  official  method  are  (1)  that  r—  =  1, 

and  (2)  that  y  =  x.  For  this  particular  month  of 
September,  r  =  .685;  a,,  =  13.80;  a,  =  9.51;  y  =  101.5; 

X  =  98.6.    These  values  make  r—  =  .994;  and  y  —  x  = 

o-x 

2.9.     The  implicit  assumptions  in  the  official  method 

are,  therefore,  in  case  of  this  one  month  of  September, 

in  close  agreement  with  the  facts,  and,  consequently, 

the  results  given  by  the  two  methods  are  almost  the 

same. 

Now  that  we  have  a  means  for  testing  the  accuracy 
of  the  official  method  of  forecasting  we  are  in  a  position 
to  answer  the  questions  which  we  asked  awhile  ago, 
one  of  which  was:  What  is  the  relative  value  of  the  of- 
ficial forecasts  from  the  data  for  May,  June,  July, 
August,  and  September?  In  Table  7  we  have  a  sum- 
mary view  of  the  important  facts. 

This  Table  reveals  a  number  of  facts  of  practical 
significance:  (1)  If  we  compare  the  values  of  S  and  S' 


74      Forecasting  the  Yield  and  the  Price  of  Cotton 

we  find  that,  in  case  of  all  of  the  months,  S  is  less 
than  S';  that  is  to  say,  the  method  of  correlation  gives 
a  more  accurate  result  than  the  method  of  the  Bureau 
of  Statistics,  although  the  difference  between  the  ac- 
curacy of  the  two  methods,  except  for  the  month  of 
May,  is  very  small.  But  this  is  the  least  important 
fact  revealed  by  the  Table. 


TABLE  7.  —  Degrees   of   Accuracy   in   the    Monthly   Official 
Forecasts  of  the  Yield  Per  Acre  of  Cotton  in  the  United 

States 


Month 

Relation  between 

actual  yield  and 

predicted  yield 

Value  of  r 

I             N              J 

S  =  a-y^l  -  r- 

May 

—  .049 

13.79 

16.05 

June 

.292 

13.20 

13.61 

July 

.595 

11.09 

11.51 

August 

.576 

11.28 

11.74 

September 

.685 

10.06 

10.47 

(2)  We  see  from  the  column  giving  the  value  of  r 
that  the  May  report  as  to  condition  and  probable 
yield  per  acre  has  no  value  whatever;  or,  since  it 
is  supposed  by  farmers,  brokers,  and  manufacturers 
to  throw  some  light  on  the  probable  yield,  we  may 
put  the  case  stronger  and  say  that  the  report  is  worse 
than  useless.  The  criticism  is  fortified  by  a  con- 
sideration of  the  value  of  S'  for  the  month  of  May, 
which  is  16.05.  This  indicates  that  the  root-mean- 
square  of  the  deviations  of  the  actual  series  from  the 
predicted  series  exceeds  the  standard  deviation  of  the 
actual  series,  which  is  tr^  =  13.80.    It  would,  therefore, 


The  Government  Crop  Reports  75 

be  much  more  profitable  to  pay  no  heed  whatever  to 
the  May  report,  which  is  issued  about  the  first  of  June, 
and  to  assume  that  the  mean  value  of  the  actual  series 
for  the  past  twenty-five  years,  namely,  101.5,  will  be 
the  probable  ratio  of  ^lYh-  For  if  one  followed  the 
official  report  on  condition  and  prospective  yield,  one's 
error  would  be  measured  by  S'  =  16.05,  whereas  if  no 
attention  were  paid  to  the  report,  the  error  would  be 
a,  =  13.80. 

(3)  The  Table  indicates  further  that  the  report  for 
the  month  of  June,  which  is  issued  about  the  first  of 
July,  has  very  little  value.  The  correlation  between 
the  actual  series  and  the  predicted  series  is  r  =  ,292, 
and  the  value  of  *S  =  13.20,  while  S'  =  13.61.  Since 
the  value  of  ay  is  13.80,  we  are  justified  in  saying  that 
the  report  for  June  has  very  little,  if  any,  value  as  a 
basis  for  predicting  the  probable  yield. 

(4)  Another  fault  revealed  by  the  Table  is  that  the 
July  forecast  is  at  least  as  accurate  as  the  August  fore- 
cast. This  is  shown  by  the  value  of  r  for  July  exceeding 
the  value  of  r  for  August,  and  by  the  values  of  S  and  S' 
for  July  being  less  than  the  corresponding  values  for 
August.  As  far  as  these  two  months  are  concerned  it 
is  not  true  that  as  the  harvest  approaches  the  reports 
are  increasingly  accurate. 

Our  method  of  testing  the  accuracy  of  the  official 
forecasts  brings  out  another  fault  of  practical  impor- 
tance and  of  theoretical  interest.  We  have  agreed  to 
measure  the  scatter  of  the  actual  yield-ratio  about  the 

r(2(F/f,_  C/^J 
predicted  yield-ratio  by  »S'  =  y    j '^ — 

As  in  computing  this  value  one  subtracts  for  each  year 


76      Forecasting  the  Yield  and  the  Price  of  Cotton 

C/q^  from  Y/y^,  it  is  possible  to  see  how  often  the  pre- 
dicted yield-ratio  exceeds  or  falls  short  of  the  actual 
yield-ratio.  The  question  is  of  importance  because, 
if  the  forecasts  show  a  tendency  to  fall  short  of  the 
actual  yield,  the  effect  of  the  forecasts  will  be  to  create 
an  undue  rise  in  price  and  thereby  favor  the  producers. 
The  opposite  result  would  be  the  case  if  the  forecasts 
show  a  tendency  to  exceed  the  actual  yield.  The  record 
of  twenty-five  years  —  in  Table  8  —  shows  that  there 
has  been  a  tendency  of  the  forecasts  of  each  month  to 
favor  the  producer.  In  the  report  of  the  condition  of 
the  crop  for  the  month  of  May,  the  official  method 
gives  a  forecast  falhng  short  of  the  actual  yield-ratio, 
19  times  out  of  25;  for  the  month  of  June,  16  times  out 
of  25;  for  the  month  of  July,  15  times  out  of  25;  for 
August,  16  times  out  of  25;  for  September,  15  times  out 
of  25.  These  figures  undoubtedly  show  a  tendency  in 
the  official  method  of  forecasting  to  give  an  under- 
estimate of  the  probable  yield. 

The  cause  of  this  bias  in  the  method  does  not  lie  on 
the  surface.  It  might  be  supposed  that  the  cause  is 
due  to  the  tendency  of  the  farmer  correspondents  of 
the  Department  of  Agriculture  to  take  a  pessimistic 
view  of  the  agricultural  outlook.  But  such  an  imputa- 
tion of  pessimism  to  the  farmer  is  not  warranted  by 
the  experience  of  the  Bureau  of  Statistics.  In  the  Crop 
Reporter,  the  official  "medium  of  communication  be- 
tween the  Division  of  Statistics  and  the  crop  reporters 
of  the  Department  of  Agriculture,"  we  are  told: 

"...  Correspondents  are  frequently  reminded" 
that  "there  has  always  been  a  tendency  to  overesti- 
mate the  average  yield  per  acre.     This  is  accounted 


The  Government  Crop  Reports 


77 


TABLE  8.  —  The  Tendency  of  the  Official  Method  op  Fore- 
casting TO  Underestimate  the  Yield  Per  Acre  of  Cotton 
IN  THE  United  States 


Year 

Value  of  (^/p^-%J 

May 

June 

July 

August 

September 

1890 

+    8.6 

+    6.0 

+    6.8 

+  5.6 

+    5.3 

1 

+    4.7 

+    1.3 

—    0.8 

+    1.9 

+    4.5 

2 

+  19.1 

+  19.1 

+  23.6 

+  24.3 

+  22.0 

3 

—  18.7 

—  14.1 

—  12.2 

—    8.6 

—  11.1 

4 

+    4.2 

+    3.8 

—    0.3 

+    0.3 

—    2.2 

5 

—    8.3 

—    8.8 

—    5.1 

—    2.6 

—    0.2 

6 

—  43.8 

—  37.4 

—  24.8 

—  12.2 

—  12.4 

7 

+  14.3 

+  10.5 

+    4.3 

+    4.1 

+  10.3 

8 

+  34.3 

+  31.2 

+  27.1 

+  29.4 

+  28.5 

9 

+    7.9 

+    6.1 

+    7.4 

+  15.1 

+  17.4 

1900 

+  17.9 

+  26.3 

+  21.9 

+  18.1 

+  12.0 

1 

+    0.5 

0.0 

+    1.2 

—    5.9 

+    2.0 

2 

—  13.3 

—    1.0 

+    0.8 

+  12.0 

+  12.6 

3 

+    6.0 

—    0.3 

—    5.7 

—  23.9 

—    8.9 

4 

+  13.4 

+    4.3 

—    2.3 

—    6.5 

—    8.2 

5 

+    7.1 

+    5.2 

+    7.8 

+    2.2 

—    8.8 

6 

+    6.8 

+    7.6 

+    7.5 

.+    6.1 

+    1.0 

7 

+    8.1 

+    5.4 

+    2.0 

—    2.8 

—    5.8 

S 

+    0.7 

+    1.0 

+    0.3 

+    4.8 

+    3.9 

9 

—  22.9 

—  13.1 

—    8.4 

—    3.5 

—    2.4 

1910 

—  11.1 

—  10.8 

—    4.2 

—    6.4 

—    4.1 

11 

+    5.0 

+    2.8 

+    0.6 

+  14.2 

+    8.7 

12 

+    7.0 

+    4.0 

+    8.4 

+    0.9 

+    0.9 

13 

+    2.5 

—    1.9 

—    1.4 

+    4.4 

+    3.4 

14 

+  24.0 

+  16 . 6 

+  17.5 

+    4.0 

+    3.1 

for  by  local  pride,  by  the  publicity  that  is  given  to 
large  individual  yields,  and  by  forgetfulness  of  the  fact 
that  there  is  in  every  agricultural  community  a  large 
number  of  farms  on  which,  during  even  the  most  favor- 


78      Forecasting  the  Yield  and  the  Price  of  Cotton 

able  seasons,  the  yield  of  the  particular  crop  is  small."  ^ 
{Crop  Reporter,  November,  1899,  p.  1.) 

Moreover,  even  if  the  farmer  could  be  proved  to 
take  either  a  pessimistic  or  an  optimistic  view  of  the 
outcome  of  the  crops,  the  effects  of  his  bias  would  be 
eliminated  in  the  official  method  of  forecasting.  The 
official  formula  is  ^VFs  =  ^ICi^  ^^^  ^^^  numerator 
and  the  denominator  in  the  fraction  CJCo  would  be 
equally  affected  by  the  farmer's  bias;  and,  consequently, 
its  influence  would  be  eliminated  in  the  ratio.  The  same 
thing  would  be  true  if  there  were  a  tendency  in  the 
Crop-Reporting  Board  either  to  overestimate  or  to 
underestimate  the  condition  of  the  growing  crop. 

The  fault  lies  with  the  method  of  forecasting  and  not 
with  the  farmers,  or  with  the  Crop-Reporting  Board. 
Let  us  recall  what  was  said  awhile  ago  about  the  im- 
plicit assumptions  in  the  official  method.  The  official 
formula  is  Y/y,^  =  C'/^^.^  and  we  have  written  this  in 
the  form  y  =  x,  where  y  =  Y/f^,  and  x  =  C/c^-  The 
ofl&cial  formula  assumes  that  the  relation  between  y 
and  X  is  linear,  but  we  know  that  the  general  formula 

^  The  purport  of  the  above  quotation  seems  to  be  contradicted  by  a 
later  official  statement.  In  the  Crop  Reporter  for  January,  1905,  there 
is  an  account  of  a  "Hearing  Before  the  Committee  of  Agriculture  of 
the  House  of  Representatives,"  relating  to  the  methods  of  estimating 
the  acreage,  production,  and  yield  per  acre  of  cotton.  Mr.  John  Hyde, 
at  that  time  Chief  of  the  Bureau  of  Statistics  of  the  Department  of 
Agriculture,  testified,  in  part,  as  follows: 

"The  Chairman.  Right  there,  let  me  ask  you  a  question.  Do  you 
find  that  your  sources  of  information,  as  a  rule,  are  prone  to  under- 
estimate? 

"Mr.  Hyde.  They  always  have  been  and,  as  a  consequence,  the 
Department  has  never  overestimated  a  crop. 

"The  Chairman.  Your  correspondents  that  bring  the  information 
in  to  you  are  prone  to  underestimate? 

"Mr.  Hyde.  With  very  few  exceptions;  yes,  sir." 


The  Government  Crop  Reports 


79 


for  a  linear  relation  between  y  and  x  is  {y  ~  y)  =  r 
—  {x  —  x).      Since  the  official  method  assumes  that 


y 


=  X,  it  implicitly  assumes  (1)  that  r  —  =  1,  and  (2) 

y  —  X  =  0.  It  takes  no  account  whatever  of  the  degree 
of  correlation  between  the  two  series,  nor  of  the  relative 
variabilities  of  the  two  series,  nor  of  the  difference  in 
the  mean  values  of  the  two  series;  and  here  lies  the 
explanation  of  the  tendency  of  the  official  method  to 
give  an  underestimate  of  the  yield  per  acre  of  cotton. 

TABLE  9.  — •  The  Implicit  Assumptions  in  the  Official  Method 
OP  Forecasting  Are:  (1)  r —  =  1;  (2)  y — •  x  =  0 


Month 

;• 

<Ty 

o-x 

u 

X 

y  —  X 

May 

—  .049 

13.80 

6.45 

—  .105 

101.5 

98.5 

3.0 

June 

.292 

13.80 

6.15 

.655 

101.5 

99.0 

2.5 

July 

.595 

13.80 

7.17 

1.145 

101.5 

98.6 

2.9 

August 

.576 

13.80 

9.18 

.866 

101.5 

98.5 

3.0 

September 

.(385 

13.80 

9.51 

.994 

101.5 

98.6 

2.9 

The  accompanying  Table  9  shows  how  very  far  from 
agreement  with  the  facts  are  the  implicit  assumptions 
in  the  official  method.  For  all  five  of  the  months  be- 
tween seeding  and  harvest,  the  mean  of  the  actual 
yield-ratio  exceeds  the  mean  of  the  predicted  yield- 
ratio;  that  is  to  say,  y  is  greater  than  x;  and  in  none  of 

the  months,  except  September,  is  the  value  of  r  —  ap- 


80      Forecasting  the  Yield  and  the  Price  of  Cotton 

proximately  equal  to  unity.  The  figures  for  September 
afford  the  simplest  illustration  of  the  inherent  bias 

in  the  official  method.    Since  r  —  is  approximately  equal 

to  unity  for  the  month  of  September,  the  general  equa- 
tion to  the  line  connecting  y  with  x,  namely,  {y  —  y)  = 

r  —  {x  —  x),  may  be  written   {y  —  y)  =  {x  —  x);  or, 

transposing  the  y,  we  may  write  the  equation  in  the 
form  y  =  x  -\-  {y  —  x).  The  official  method  assumes 
that  {y  ~  x)  is  equal  to  zero,  when,  according  to  the 
actual  figures  for  the  month  of  September,  {y  —  x)  = 
2.9.  The  equation  to  the  straight  line  ought  therefore 
to  be  ?/  =  X  +  2.9,  whereas  the  official  formula  is  ?/  = 
X,  and,  consequently,  it  gives  a  forecast  of  the  value 
of  y  which,  on  the  average,  falls  short  of  the  true 
value  by  2.9. 

One  of  the  questions  naturally  suggested  with  refer- 
ence to  the  official  method  of  forecasting  is  whether  a 
four  years  progressive  average,  or  a  three  years  pro- 
gressive average,  would  not  give  as  good  results,  when 
used  as  a  basis  of  forecasting,  as  the  five  years  pro- 
gressive average  that  is  adopted  by  the  Bureau  of 
Statistics.  The  device  that  we  have  used  to  test  the 
accuracy  of  the  official  forecasts  enables  us  to  obtain 
definite  information.  The  results  of  the  necessary 
computations  are  given  in  Tables  10  and  11. 

The  findings  in  Table  10  seem  to  justify  the  following 
conclusions : 

(1)  A  comparison  of  the  values  of  S  and  S',  in  each 
of  the  main  divisions  of  the  Table,  shows  that,  in  all  of 
the  fifteen  cases,  the  value  of  S  is  less  than  the  value 


The  Government  Crop  Reports 


81 


TABLE  10.  —  Results  of  Forecasting  the  Yield  Per 
Acre  of  Cotton  by  the  Method  of  Progressive 
Averages 


Month 

Progressive  Average  of  Five  Years 

Relation 
between 

actual 

yield  and 

forecast 

yield 

Value  of  r 

Scatter  about 
the  line  of 
regression 

0-,,  =  13.80 

Scatter  about   the  hne  of 
progressive   averages 

^       I                  N                  J 

May 

—  .049 

13.79 

16.05 

June 

.292 

13.20 

13.61 

July 

.595 

11.09 

11.51 

August 

.576 

11.28 

11.74 

September 

.  685 

10.06 

10.47 

.Montli 

Progressive  Average  of  Four  Years 

Relation 
between 

actual 

yield  and 

forecast 

yield 

Value  of  r 

Scatter  about 
the  line  of 
regression 

'''' =  0",;/v'l -'•■-' 
or  ,^  =  13.84 

Scatter  about  the  line  of 
progressive  averages 

I                .V                1 

May 

—  .073 

13.80 

16.12 

June 

271 

13.32 

13.45 

July 

.576 

11.31 

11.58 

August 

.564 

11.43 

11.77 

September 

.683 

10.11 

10.42 

Month 

Progressive  Averagi-  of  Three  Years 

Relation 

between 

actual 

yield  and 

forecast 

yield 

Value  of  r 

Scatter  about 
the  line  of 
regres.sion 

0-,/=14..-.i 

Scatter  about  the  line  of 
progressive  averages 

^       I                  A'                  1 

May 

.059 

14.51 

16.04 

June 

.380 

13.45 

13.61 

July 

.650 

11.03 

11.35 

August 

.599 

11.64 

11.86 

September 

.724 

10.03 

10.26 

82       Forecasting  the  Yield  and  the  Price  of  Cotton 

of  S';  but  that  the  difference  is  always  small  and  theo- 
retically insignificant,  except  for  the  month  of  May.  For 
that  one  month,  whether  one  employs  the  official  for- 
mula with  a  five,  four,  or  three  years  progressive  mean, 
the  value  of  *S',  since  in  each  case  it  exceeds  the  cor- 
responding value  of  (7,^,  shows  that  the  forecast  is  worse 
than  useless; 

(2)  When  attention  is  paid  to  the  probable  errors  of 
8  and  S>' ,  there  is  really  no  difference  in  the  accuracy 
of  the  forecasts  whether  they  are  based  upon  a  five, 
four,  or  three  years  progressive  average; 

The  findings  in  Table  11  enable  us  to  add  to  the  fore- 
going, 

(3)  The  disposition  of  the  official  formula  to  give  an 
underestimate  of  the  yield  per  acre  is  shown,  no  matter 
whether  the  progressive  average  is  one  of  three,  four, 
or  five  years;  but  the  bias  is  perhaps  less  in  case  of  the 
three  years  average  than  in  case  of  the  five  years  aver- 
age; 

(4)  The  correlation  prediction  formula  shows  no  bias. 

Our  general  conclusion  is  that  because  of  its  greater 
accuracy  and  freedom  from  bias  the  correlation  for- 
mula, with  either  a  three  or  five  years  progressive 
average,  is  preferable  to  the  official  formula  with  the 
five  years  progressive  average. 

Acreage  and  Production 

The  two  factors  that  are  used  to  estimate  the  prob- 
able size  of  the  cotton  crop  are  the  probable  yield  per 
acre  and  the  number  of  acres  under  cultivation.  The 
problem  of  forecasting  the  probable  yield  per  acre  has 


The  Government  Crop  Reports 


83 


Q 

o 

« 
Ph 
H 
35 
H 


K 

o 

o 
<; 

o 

o 


o 


H     a 


1 

CD 

o 

,_( 

-*l 

00 

CD 

CO 

o 

'^ 

"^^     .1.2 

OS 

lO 

lO 

d 

'^J* 

c 

C3 
c 

E 

00 

+ 

+ 

7 

+ 

1 

■2    II     g  2 

H 

w 

H 

H 

H 

P 

r^ 

t^ 

I> 

IM 

CD 

.« 

o      cj       =J 

I— 1 

CD 

l^ 

Ci 

CD 

m 

T—H 

00 

r— H 

O 

t-      fc    J3 

2 

l-      a     -^ 

.ai  II 

II 

II 

II 

II 

II 

p. 

a 

=r: 

as 

»J 

S5 

2;; 

S.2  i.2 

£§i 

ro 

fO 

o 

C<l 

C<1 

>1 

£ 

a""! 

J3 

a)       o 

"S  2 

r^ 

^ 

(M 

'^ 

rr\ 

SEE 

'-' 

"^ 

'-' 

'-' 

"-• 

z|g^ 

0£ 

<u 

■* 

•* 

O 

■n^ 

lO 

t- 

CO 

(M 

CD 

lO 

X    —     o 

^   "  'Zl 

LO 

!M 

CO 

CD 

03      II       S 

^ 

:r 

1— 1 

3    a  -c 

E  -a  :§ 

+ 

+ 

1 

+ 

+ 

s 

o  i  ^ 

H 

H 

H 

H 

H 

"*■         .i-TS 

CO 

CD 

iM 

(N 

^ 

S    lO  'u 

■* 

05 

CD 

CD 

O 

"S 

2  ^-C!*^ 

lO 

O 

00 

T-H 

1 

5^    ^ 

l' 

.— 1 

tc 

£   II    ^ 

1 

£ 

1   «   " 

II 

II 

II 

11 

II 

c 

Sj 

^ 

=5 

as 

=!5 

^■2  £.2 

C3        =5 

C2 

E  "S  a"S 

^  2  s 

(M 

ro 

o 

•M 

^ 

>< 

'.^    V    QJ    V* 

^ 

1-H 

t« 

>»h-3j:'^ 

o"  C 

o 

°!2^!2 

O     £ 

g  2  g  o 

3  .Z-^  :j  o 

3| 

t^ 

CD 

(M 

lO 

CO 

|i 

1— 1 

^ 

T— ( 

O  o 

Cj 

Tf 

I— 

TT 

,—( 

o 

S    2    - 

00 

CD 

rti 

(N 

TJH 

-g   ~   .^ 

^ 

CD 

_i 

CD 

CO 

"^  II  1 

OD 

y—^ 

c3 

3    s>  2 

+ 

+ 

1 

+ 

+ 

E 

£    §  '^ 

H 

H 

H 

H 

H 

^               .o-^ 

lO 

IC 

lO 

CD 

Tj< 

> 

.2    '^- 

o 

lO 

■* 

CD 

O) 

1— 1 

CD 

00 

O^ 

■£  II    ^ 

t 

■^ 

0 

6  ^ 

11 

II 

11 

11 

II 

Ss 

Si 

S: 

=n 

»5 

g.2  i.2 

o!         =« 

oi 

S  "S    Q."^ 

■3  S  E 

CO 

-v^ 

,_, 

(M 

C<J 

>. 

*^     1     OJ     ' 

fc-2  E 

^ 

1— t 

t-H 

> 

o   a 

isl 

05 

CD 

lO 

CD 

lO 

E  S  o  ^ 

|i 

t-H 

T— * 

T-H 

3  *^  =j  t; 

Z    S    OJ^ 

o^ 

(H 

+J 

Ol 

5 

o3 

^ 

CO 

g 

3 

§ 

3 

►^ 

< 

S 
a 

a 

84       Forecasting  the  Yield  and  the  Price  of  Cotton 

already  been  dealt  with,  and  only  a  few  words  need  be 
said  about  the  official  method  of  estimating  acreage. 

Throughout  the  whole  period  under  investigation, 
1890-1914,  the  Bureau  of  Statistics  has  again  and 
again  warned  the  public  that  its  figures  referring  to 
acreage  are  merely  estimates  and  not  the  results  of 
extensive  measurements  such  as  are  used  by  the  Bureau 
of  the  Census.  It  has  issued  its  reports  as  ''the  best 
available  data,  representing  the  fullest  information 
obtainable  at  the  time  they  are  made,"  ^  and  it  has 
frankly  pointed  out  the  limitations  of  the  method 
which  it  has  felt  compelled  to  follow  in  estimating 
acreage. 

The  census  figures  of  the  acreage  devoted  to  the 
several  crops,  which  have  appeared  every  ten  years, 
have  been  taken  by  the  Bureau  of  Statistics  as  the 
foundation  upon  which  to  base  its  calculation  as  to  the 
acreage  under  cultivation  during  the  intercensal  years. 
For  each  year  between  the  census  surveys,  correspond- 
ents were  asked  to  observe  whether,  as  compared  with 
the  preceding  year,  there  had  been  an  increase  or  a 
decrease  in  the  acreage  of  cotton  in  their  respective 
districts,  and  to  express  the  change  as  a  percentage 
change.  The  Bureau  of  Statistics,  using  the  last  re- 
turns of  the  Census  as  the  best  available  data,  has 
computed  the  absolute  value  of  the  combined  per- 
centages of  its  correspondents  and  has  issued  the  result 
as  the  Department's  estimate  of  the  acreage  of  the 
cotton  crop  for  the  current  year.  The  Bureau  has  re- 
garded each  of  its  estimates  merely  as  "a  consensus  of 

1  Annual  Report  of  the  Bureau  of  Statistics  for  the  Fiscal  Year  1911- 
1912.    Crop  Reporter,  December.  1912. 


The  Government  Crop  Reports  85 

judgment  of  many  thousands  of  correspondents,"  ^  and 
it  has  pointed  out  ''that  estimates  made  monthly  from 
year  to  year,  following  each  other  during  a  period  of  10 
years,  without  means  of  verification  or  correction,  are 
likely  to  be  more  or  less  out  of  line  with  conditions  at 
the  end  of  the  10-year  period  as  disclosed  by  actual 
census  enumerations.  Cumulative  errors,  impossible  of 
discovery,  are  likely  to  occur  and  cannot  be  corrected 
until  census  reports  are  available.' '  -  At  the  appearance 
of  new  census  figures  the  Bureau  of  Statistics  has 
revised  its  estimates  of  the  preceding  intercensal 
years,  and  the  more  recent  census  figures  have  been 
used  as  the  basis  of  estimates  for  the  follo\\dng 
years. 

It  would  doubtless  be  possible  to  test  the  degree  of 
accuracy  in  the  method  employed  by  the  Bureau  of 
Statistics  for  calculating  the  acreage  of  the  crops;  or,  to 
be  more  exact,  it  would  be  possible  to  test  how  nearly 
the  preliminary  estimates  correspond  with  the  revised 
estimates.  As  far  as  I  am  aware  this  test  has  never 
been  carried  out.  The  Bureau  reports  that,  in  case  of 
some  of  the  crops,  there  has  been  a  considerable  differ- 
ence in  the  two  estimates.^  If  the  test  were  made  and 
the  method  were  found  to  be  unsatisfactory,  the  prob- 
lem would  then  present  itself  of  finding  a  better  method, 
and  the  solution  of  the  problem  would  be  sought  in 
either  of  two  directions:  Either  the  direct  measure- 
ment, such  as  is  used  bj^  the  Bureau  of  the  Census,  must 

^  Annual  Report  of  the   Bureau  of  Statistics  for  the  Fiscal  Year 
1911-1912. 
^  Ibidem. 

^  Crop  Reporter,  May,  1900,  p.  2.     " Dei)artment  of  Agriculture  and 
he  Census." 


86       Forecasting  the  Yield  and  the  Price  of  Cotton 

be  applied  more  frequently  than  ten  years,  and  the 
method  of  estimates  employed  by  the  Bureau  of 
Statistics  be  checked  up  at  shorter  intervals;  or  else 
the  quantitative  connections  between  variations  in 
acreage  and  the  variations  in  other  economic  factors 
must  be  discovered,  and  the  acreage  be  then  computed 
from  these  known  connections.  The  former  solution, 
which  is  undoubtedly  the  best,  is  urged  by  the  Depart- 
ment of  Agriculture.^  But  an  agricultural  survey  is 
extremely  expensive,  and  its  results  are  frequently 
made  known  when,  for  many  practical  purposes,  it  is 
too  late.  The  Bureau  of  Statistics  has  reported  that 
''the  results  of  the  agricultural  census  which  related  to 
1909  were  not  published  in  time  to  permit  a  revision  of 
estimates  of  this  Bureau  until  the  close  of  1911."  ^ 
Furthermore,  while  the  Bureau  of  Statistics  makes  its 
estimates  for  current  use,  the  estimate  of  the  acreage  of 
cotton  is  not  published  until  about  July  1.  But  there 
are  a  number  of  industries  dependent  upon  the  acreage 
of  cotton  which  would  profit  by  having  a  reliable 
estimate  earlier  in  the  year.  Would  it  not  be  possible 
to  have  a  fair  estimate  of  the  probable  acreage  even 
before  the  crop  is  planted? 

In  Table  12  there  is  an  illustration  of  a  method  by 
which  a  solution  may  be  obtained  of  the  problem  that 
has  just  been  described.  The  acreage  planted  in  cotton, 
any  given  year,  is  largely  dependent  upon  what  has 
been  the  fortune,  good  or  bad,  of  the  cotton  farmers  in 
preceding  years.  If,  for  example,  the  price  of  cotton  has 
been  falling,  few  acres  will  be  seeded  in  that  particular 

1  Annual  Report  of  the  Bureau  of  Statistics  for  the  Fiscal  Year  1911- 
1912. 
"  Ibidem. 


The  Government  Crop  Reports 


87 


TABLE  12.  —  Percentage  Change  in  the  Acreage  of  Cotton  and 
Percentage  Change  in  the  Production  op  Cotton  Lint 


Year 

Acreage  of 

Cotton 

(Thousands 

of  acres) 

Absolute 
change  in 
acreage 

Percentage 

change  in 

acreage 

Produc- 
tion of 
cotton  lint 
(Millions 
of  bales) 

Absolute 
change  in 
production 

Percentage 
change  in 
production 

1888 

6.92 

9 

20,180 

7.47 

+  0,55 

+    7.95 

1890 

21,886 

+     1706 

+    8.45 

8.56 

+  1,09 

+  14.59 

1 

23,876 

+    1990 

+    9.09 

8.94 

+  0,38 

+  4.44 

2 

15,228 

—    8648 

—  36.22 

6.66 

—  2.28 

—  25,50 

3 

23,837 

+    8609 

+  56.53 

7.43 

+  0,77 

+  11.56 

4 

24,959 

+    1122 

+    4.71 

10.03 

+  2,60 

+  34,99 

5 

21,896 

—    3063 

—  12.27 

7.15 

—  2,88 

—  28,71 

6 

32,823 

+  10927 

+  49.90 

8.52 

+  1.37 

+  19,16 

7 

28,861 

—    3962 

—  12.08 

10.99 

+  2.47 

+  28,99 

8 

25,174 

—    3687 

—  12.78 

11,44 

+  0,45 

+    4.10 

9 

24,278 

—      896 

—    3.56 

9.35 

—  2,09 

—  18.27 

1900 

24,982 

+      704 

+    2.90 

10.12 

+  0.77 

+    8.24 

1 

26,897 

+     1915 

+    7,67 

9.51 

—  0.61 

—    6.03 

2 
3 

26,940 

+        43 

+    0.16 

10.63 

+  1,12 

+  11.78 

26,952 

+         12 

+    0.04 

9.85 

—  0,78 

—    7.34 

4 

31,350 

+    4398 

+  16.32 

13.44 

+  3,79 

+  36.45 

5 

27,205 

—    4145 

—  13.22 

10,58 

—  2,86 

—  21.28 

6 

31,301 

+    4096 

+  15.06 

13,27 

+  2.69 

+  25.43 

7 

29,848 

—    1453 

—    4.64 

11.11 

—  2.16 

—  16.28 

8 

.32,493 

+    2645 

+    8.86 

13.24 

+  2,13 

+  19.17 

9 

31,060 

—    1433 

—    4.41 

10.00 

—  3,24 

—  24.47 

1910 

32,467 

+    1407 

+    4.53 

11.61 

+  1,61 

+  16.10 

11 

36,045 

+    3578 

+  11.02 

15,69 

+  4.08 

+  35.14 

12 

34,283 

—    1762 

—    4.89 

13,70 

—  1.99 

—  12.68 

13 

37,089 

+    2806 

+    8.18 

crop.  There  should  therefore,  in  normal  times,  be  some 
relation  between  the  percentage  change  in  the  price  of 
cotton  last  year  over  the  preceding  year  and  the  per- 
centage change  in  the  acreage  of  cotton  this  year  over 


88      Forecasting  the  Yield  and  the  Price  of  Cotton 

last  year.  In  Table  12  the  data  are  presented  with 
which  to  compute  the  relation  between  the  two  va- 
riables, namely,  the  percentage  change  in  the  acreage  of 
a  given  year  over  the  acreage  of  the  preceding  year,  and 
the  percentage  change  in  the  price  of  cotton  from  the 
price  prevailing  two  years  before  the  current  year  to 
the  price  the  year  before  the  current  year.  In  the  same 
way  that  this  correlation  Table  was  prepared,  similar 
Tables  were  compiled  connecting  the  percentage 
change  in  the  acreage  of  cotton  with  the  percentage 
change,  in  the  preceding  year,  of  other  variables.  A 
summary  of  the  calculations  is  here  given : 

The  correlation  between  the  percentage  change  in  the 
acreage  of  cotton  and 

(1)  the  percentage  change  of  the  year  before  in  the 

total   production   of   cotton   lint,   r  =  —  .641  ; 

(2)  the  percentage  change  of  the  year  before  in  the 

price  per  pound  of  cotton  Hnt,  r  =  .532; 

(3)  the  percentage  change  of  the  year  before  in  the 

value  of  the  yield  per  acre  of  cotton   Unt, 
r  =  .508; 

(4)  the  percentage  change  of  the  year  before  in  the 

acreage  of  cotton,  r  =  — .492; 

(5)  the  percentage  change  of  the  year  before  in  the 

yield  per  acre  of  cotton,  r  =  —  .217; 

(6)  the  percentage  change  of  the  year  before  in  the 

index    number    of    general    wholesale    prices, 

r  =  .005. 
From  these  calculations  it  is  clear  that  even  before 
the   cotton   crop   is   planted,   it   is   possible   to   fore- 
cast   the    probable    acreage    with    substantially    the 
same   degree    of    accuracy   with   which    the    Bureau 


The  Government  Crop  Reports  89 

of  Statistics  can  forecast  the  yield  per  acre  of  cot- 
ton at  the  first  of  September.  We  know  from  the 
results  of  the  preceding  chapter  that  when  the  cor- 
relation between  two  variables  is  linear,  the  scatter 
of  the  observations  about  the  line  of  regression  is 
measured  by  <S  =  dyS/l  —  r'-.  The  degree  of  accuracy 
with  which  we  can  forecast  results  is,  therefore,  de- 
pendent upon  the  two  factors  a-y  and  \/l  —  r'.  If  we 
make  allowance  for  the  difference  between  the  values  of 
(Ty  in  case  of  two  series,  the  relative  degree  of  accuracy 
with  which  we  can  forecast  results  is  dependent  upon 
\/l  ~  r-;  the  smaller  the  value  of  s/l  —  r',  the  better 
the  forecast.  We  have  found  that  the  correlation 
between  the  actual  yield  of  cotton  and  the  yield  as 
predicted  by  means  of  the  official  formula  is,  at  the 
first  of  August,  r  =  .595,  and  at  the  first  of  September, 
r  =  .576.  The  correlation  between  the  percentage 
change  in  the  acreage  of  cotton  for  any  given  year  and 
the  percentage  change  in  the  production  of  cotton  the 
preceding  year  is  r  =  —.641;  and  the  correlation 
between  the  percentage  change  in  the  acreage  of  any 
given  year  and  the  percentage  change  in  the  price  per 
pound  of  cotton  fint  the  preceding  year  is  r  =  .532. 
It  is  true,  therefore,  that  when  allowance  is  made  for 
the  difference  in  the  variabilities  of  the  things  com- 
pared, it  is  possible  to  forecast  the  acreage  of  cotton 
several  months  before  the  crop  is  planted,  with  as 
great  a  degree  of  accuracy  as  the  Bureau  of  Statistics 
can  forecast  the  possible  yield  per  acre  of  cotton  at  the 
first  of  September.^ 

^  It  will  be  understood,  I  hope,  that  I  am  not  offering  this  method  of 
forecasting  acreage  as  a  substitute  for  the  method  employed  by  the 
Bureau  of  Statistics.    I  present  it  for  its  practical  value  in  supplement- 


90       Forecasting  the  Yield  and  the  Price  of  Cotton 

If  we  gather  into  a  summary  the  principal  points 
that  have  been  made  in  this  chapter,  we  may  list  them 
as  follows : 

1.  By  means  of  a  vast  and  remarkable  statistical 
organization  with  thousands  of  correspondents,  paid 
and  unpaid,  the  Department  of  Agriculture  collects  its 
data  referring  to  the  condition  of  the  cotton  crop,  and 
issues  monthly  reports  during  the  period  between 
seed-time  and  harvest.  As  the  reports  have  great 
influence  upon  the  price  of  cotton,  every  precaution  is 
taken  to  prevent  any  leakage  of  information  before  the 
final  conclusions  are  given  to  the  public. 

2.  The  Bureau  of  Statistics  of  the  Department  of 
Agriculture  has  devised  a  method  of  forecasting,  from 
the  monthly  records  of  the  condition  of  the  crop,  the 
probable  yield  per  acre  of  cotton.  As,  however,  the 
raw  data  are  largely  estimates  supplied  by  its  corre- 
spondents, the  Bureau  has  regarded  the  material  upon 
which  its  estimates  are  based  as  a  "consensus  of  the 
opinion  of  the  well-informed."  Although  the  figures 
descriptive  of  the  condition  of  the  crop  are  expressed  in 
percentages  and  decimals  of  percentages,  the  Bureau  is 
aware  of  the  insecure  foundation  upon  which  its  fore- 
casts rest.  Nevertheless  the  forecasts  have  far-reaching 
effects.  Indeed,  as  the  Cotton  Belt  of  the  United 
States  produces  about  75  per  cent  of  the  world's  cotton 
crop,  the  degree  of  accuracy  in  the  work  of  the  Bureau 
of  Statistics  of  our  Department  of  Agriculture  produces 
its  effect  throughout  the  civilized  world. 

3.  When  the  cotton  reports  are  subjected  to  a  critical 

ing  the  work  of  the  Bureau  of  Statistics,  and  for  its  theoretical  value  in 
illustrating  that  our  economic  activity  is  such  a  matter  of  routine  that 
it  admits  of  prediction. 


The  Government  Crop  Reports  91 

examination  as  to  the  extent  of  their  rehabihty  and  the 
degree  of  accuracy  of  the  method  of  forecasting  the  prob- 
able yield  per  acre,  the  chief  facts  discovered  are  these : 

(a)  The  May  report  —  that  is,  the  report  referring  to 
the  condition  of  the  crop  at  the  end  of  May  —  has  no 
value  whatever  as  a  basis  upon  which  to  forecast  the 
average  yield  per  acre  of  cotton.  The  percentages  re- 
ferring to  the  condition  of  the  crop  for  the  whole  cotton 
section  are  arithmetical  means  of  wild  guesses;  any 
forecasts  based  upon  the  May  figures  are  spurious ;  and 
any  money  that  changes  hands  in  consequence  of  the 
forecasts  are  losses  and  gains  resulting  from  a  simple 
gamble ; 

(b)  The  June  report  —  which  is  issued  about  the 
first  of  July  —  as  far  as  it  refers  to  the  average  condition 
of  the  cotton  crop  in  the  whole  country,  has  a  meas- 
urable, but  small,  value  as  a  basis  of  forecasting  the 
ultimate  average  yield  per  acre.  When,  for  a  period 
extending  over  a  quarter  of  a  century,  the  degree  of 
relation  between  the  predicted  yield  and  the  actual 
yield  is  properly  measured,  it  is  found  that  the  co- 
efficient of  correlation  between  the  two  series,  the 
forecast  series  and  the  actual  series,  is  r  =  .292; 

(c)  The  July,  August,  and  September  reports  have  a 
decided  value  as  bases  of  forecasting  the  average  yield 
per  acre.  The  coefficients  measuring  the  closeness  of 
the  relation  between  the  predicted  series  and  the  actual 
series  are,  for  July,  r  =  .595;  for  August,  r  =  .576;  for 
September,  r  =  .685.  In  this  chapter  and  in  chapter  II, 
methods  are  described  by  which  the  degree  of  reliability 
of  these  reports  as  bases  of  forecasting  the  ultimate 
yield  per  acre  are  measured; 


92      Forecasting  the  Yield  and  the  Price  of  Cotton 

(d)  The  method  of  the  Bureau  of  Statistics  by  which, 
from  the  reports  on  the  condition  of  the  crop,  forecasts 
are  made  as  to  the  ultimate  yield  per  acre,  has  in- 
herent defects  that  lead  to  an  underestimate  of  the 
yield  per  acre,  and  thereby  favors  the  producers.  The 
official  method  of  forecasting,  if  applied  to  the  data 
referring  to  the  condition  of  the  crop,  during  a  period  of 
twenty-five  years,  gives  a  predicted  yield  per  acre 
which,  out  of  a  total  of  25  years,  is  an  underestimate 
19  times  when  based  upon  the  May  condition;  16  times 
when  based  upon  the  June  condition;  15  times,  in  case 
of  July;  16,  in  case  of  August;  and  15,  in  case  of  Septem- 
ber; 

(e)  It  is  possible  to  construct  a  better  prediction 
formula  than  the  one  used  by  the  Bureau  of  Statis- 
tics —  a  formula  more  accurate  in  its  forecasts  and 
entirely  free  from  any  disposition  either  to  under- 
estimate or  to  overestimate  the  yield  per  acre. 

4.  The  official  estimate  of  the  acreage  planted  in 
cotton  is  published  about  July  1,  and  the  two  factors 
in  estimating  the  ultimate  production  are  the  acreage 
and  the  yield  per  acre.  By  means  of  a  method  de- 
scribed in  this  chapter,  it  is  possible  to  forecast  the 
acreage,  several  months  before  the  crop  is  planted,  with 
a  degree  of  precision  as  great  as  that  of  the  official  fore- 
cast of  the  yield,  after  the  crop  has  completed  its  growth 
and  is  about  to  be  harvested. 


CHAPTER  IV 

FORECASTING  THE  YIELD  OF  COTTON  FROM  WEATHER 

REPORTS 

"The  essence  of  science  .  .  .  consists  in  inferring  antecedent  con- 
ditions, and  anticipating  future  evolutions,  from  phenomena  which 
have  actually  come  under  observation." 

—  Lord  Kelvin. 

In  the  preceding  chapter  the  methods  and  results  of 
the  Department  of  Agriculture  were  examined  with 
reference  to  the  accuracy  of  the  method  of  forecasting 
and  the  degree  of  value  of  the  crop  reports  as  forecasts 
of  the  yield  per  acre  of  cotton.  The  inquiry  was  con- 
cerned with  the  official  statistics  bearing  upon  the 
monthly  condition  of  the  growing  crop  and  upon  the 
annual  yield  per  acre  of  cotton  in  the  whole  Cotton 
Belt.  Holding  the  results  of  this  inquiry  in  mind,  we 
shall  consider,  in  the  present  chapter,  the  possibility 
of  forecasting  the  yield  per  acre  of  cotton  simply  from 
the  current  reports  of  the  Weather  Bureau  as  to  rainfall 
and  temperature  in  the  several  cotton  states.  Inas- 
much as  the  investigation  entails  in  case  of  each  state 
a  considerable  amount  of  statistical  computation,  the 
inquiry  will  not  be  extended  to  the  whole  area  of  the 
Cotton  Belt,  but  will  be  limited  to  a  few  representative 
states.  Remembering  always  that  our  conclusions  are 
based  upon  the  detailed  investigation  of  the  representa- 
tive states  we  shall  maintain  these  theses : 

(1)  That  some  of  the  official  reports  referring  to  the 
individual  states  are  valuable  as  forecasts,  but  that 


94      Forecasting  the  Yield  and  the  Price  of  Cotton 

others  are  worse  than  useless  in  the  sense  of  supplying 
erroneous  instruction  as  to  the  crop  outlook,  and 
thereby  suggesting  a  misdirection  of  activity  on  the 
part  of  farmers,  dealers,  and  manufacturers; 

(2)  That  even  in  case  of  the  useful  forecasts,  the 
official  method  does  not  extract  the  full  amount  of 
truth  contained  in  the  laboriously  collected  data; 

(3)  That  notwithstanding  the  vast  official  organiza- 
tion for  collecting  and  reducing  data  bearing  upon  the 
condition  of  the  growing  crop,  it  is  possible,  by  means 
of  mathematical  methods,  to  make  more  accurate 
forecasts  than  the  official  reports,  in  the  matter  of  the 
prospective  yield  per  acre  of  cotton,  simply  from  the 
data  supplied  by  the  Weather  Bureau  as  to  the  current 
records  of  rainfall  and  temperature  in  the  respective 
cotton  states; 

(4)  That  the  principles  and  methods  of  forecasting 
the  yield  of  cotton  may  be  utilized  in  forecasting  the 
yield  of  other  agricultural  crops.  ^ 

The   Official  Forecasts   of  the    Yield   of  Representative 

States 

In  1914  the  total  production  of  cotton  in  the  United 
States  was  16,135,000  bales  of  500  pounds  gross  weight. 
This  total  product,  with  the  exception  of  a  negligibly 
small  amount  produced  in  other  parts  of  the  country, 
was  the  yield  of  thirteen  states  in  the  Cotton  Belt.  The 
four  states  of  largest  yield  were  Texas,  with  its  4,592,000 
bales;  Georgia,  with  2,718,000  bales;  Alabama,  with 

1  This  thesis  will  not  be  developed  in  the  present  Essay.  The  prob- 
ability of  its  truth  is  suggested  by  the  researches  in  Chapter  III  of 
Economic  Cycles:  Their  Law  and  Cause. 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports     95 

1,751,000  bales;  South  Carolina,  with  1,534,000  bales. 
In  all,  these  four  states  produced  10,595,000  bales  or 
sixty-five  per  cent  of  the  whole  crop.^  Furthermore, 
Texas,  with  its  enormous  yield,  is  representative  of  the 
conditions  of  production  in  the  extreme  Southwest; 
Georgia  and  South  Carolina  exemplify  the  conditions 
at  the  other  extreme  of  the  Cotton  Belt  on  the  Atlantic 
Coast;  Alabama  typifies  the  conditions  on  the  Gulf  of 
Mexico.  These  four  states,  which  are  in  direct  order  of 
their  importance,  happen  to  illustrate  the  weather  con- 
ditions in  the  cotton-growing  section  and  are,  for  the 
purpose  of  our  inquiry,  representative  of  the  conditions 
of  production  in  the  whole  of  the  Cotton  Belt. 

In  order  to  measure  the  degree  of  accuracy  of  the 
official  method  of  forecasting  the  yield  per  acre  of  cotton 
in  these  representative  states,  we  shall  recall  the  de- 
scription of  the  forecasting  formula  which  was  given  in 
the  preceding  chapter,  together  with  the  description  of 
the  coefficient  measuring  the  degree  of  accuracy  of  the 
formula  when  it  is  actually  applied  to  the  given  data. 
In  symbolic  terms  the  forecasting  formula  is  ^IC-,  = 
■^/Foj  where  C  is  the  condition  of  the  cotton  crop  in  the 
current  month;  C^  is  the  mean  condition  of  the  crop  in 
the  same  month  during  the  five  years  preceding  the 
given  year;  Y  is  the  prospective  yield  per  acre  of  the 
present  year;  and  F^  is  the  mean  yield  per  acre  during 
the  preceding  five  years.  The  value  of  ^ICf,  is  called  the 
condition-ratio,  or  the  forecasting  ratio,  and  ^/Fs  is 
called  the  yield-ratio.  We  make  a  distinction  between 
the  theoretical  yield-ratio  and  the  actual  yield-ratio. 
When  in  ^/Fo,  F  is  the  predicted  value  of  the  yield  per 

1  Yearbook  of  the  Deparlment  of  Agriculture,  1915,  p.  470. 


96      Forecasting  the  Yield  and  the  Price  of  Cotton 

acre,  then  ^/Fj  is  called  the  theoretical  yield-ratio; 
and  when  Y  is  the  actual  value  of  the  jdeld  per  acre, 
y/fb  is  called  the  actual  yield-ratio.  Theoretically,  if 
the  forecasting  formula  were  perfectly  accurate,  the 
actual  yield-ratio  ^/Fs  should  always  be  equal  to 
C/^5,  but,  as  a  matter  of  record,  the  actual  yield-ratios 
differ  from  the  condition-ratios,  and  a  measure  of  the 
degree  of  accuracy  of  the  forecasting  formula,  must, 
obviously,  be  some  function  of  the  difference  between 
the  actual  yield-ratios  and  the  theoretical  yield-ratios; 
that  is  to  say,  the  coefficient  measuring  the  accuracy  of 
the  official  formula  must  be  some  function  of  (^/Fs  — 
^/Cb)-  For  reasons  that  are  fully  set  forth  in  the 
preceding  chapter  we  have  taken  as  the  coefficient 
measuring  the  degree  of  accuracy  of  the  official  formula 

N 
N  equals  the  number  of  observations. 

With  this  understanding  of  the  formula  which  we 
shall  employ,  we  shall  now  give  the  results  of  the  test 
of  the  degree  of  accuracy  with  which  the  official  fore- 
casting formula  would  enable  one  to  predict  the  actual 
yield  ratio  during  the  21  years  from  1894  to  1914.  The 
results  are  collected  in  the  accompanying  Table  13. 

In  the  preceding  chapter  we  showed  why  the  official 
formula  has  increasing  accuracy  as  the  value  of  S'  be- 
comes smaller  and  smaller.  The  object  of  any  forecast- 
ing of  phenomena  is,  of  course,  to  get  as  accurate  an 
appreciation  as  possible  of  the  phenomena. 

*S'  represents  the  degree  of  approximation  of  the  actual 
yield-ratios  to  the  predicted  yield-ratios,  and  a,,  gives 
the  variability  of  the  actual  yield-ratios  for  the  entire 


the  value  of  S'  where  S'  =  V    <  -^ — ^-^ ^-^^  r  ,  and 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports    97 

period  covered  in  the  investigation,  which,  in  this  case, 
is  the  21  years  from  1894  to  1914. 

TABLE  13.  —  Tests  of  the  Accuracy  of  the  Official  Formula 
THAT  Is  Used  in  Forecasting  the  Probable  Yield  Per  Acre 
of  Cotton  from  the  Monthly  Condition  of  the  Crop 


The  Representative 
States 

The 

Standard 
Devia- 
tion of 

the  Yield 
Ratio 

The  Accuracy  of  the  Forecasting  Formula 

May 

June 

July 

August 

Septem- 
ber 

Texas 

24.64 

26.38 

22.11 

19.23 

17.86 

13.77 

Georgia 

13.89 

17.28 

15.20 

12.17 

11.92 

11.08 

Alabama 

13.37 

17.59 

13.58 

12.24 

11.66 

10.21 

South  Carolina 

18.65 

21.90 

19.06 

17.02 

19.03 

15.28 

If,  now,  we  examine  the  results  that  are  collected 
in  Table  13,  we  see  that  the  May  report  as  to  the  condi- 
tion of  the  crop,  which  is  issued  about  the  first  of  June, 
not  only  has  no  value  in  case  of  all  four  of  the  represent- 
ative states,  but  that  it  is  worse  than  useless  since  in 
all  four  cases  the  value  of  *S'  is  greater  than  a^.  The 
forecasts  by  means  of  the  official  formula  miss  the 
actual  yield-ratios  by  an  amount  S'  which  exceeds  o-y, 
the  value  of  the  variability  of  the  actual  yield-ratios 
when  no  forecast  is  made  at  all.  The  June  reports, 
which  are  issued  about  the  first  of  July,  are  in  three  out 
of  four  cases  worse  than  useless,  because  in  case  of 
three  of  the  states  S'  is  greater  than  o-^.  The  reports 
for  July,  August,  and  September  have  real  value  as 
forecasts  and  the  value  increases  as  the  crop  approaches 
maturity. 


98      Forecasting  the  Yield  and  the  Price  of  Cotton 

In  the  preceding  chapter  it  was  made  clear  that  an 
improvement  upon  the  official  results  would  be  ob- 
tained if,  as  a  forecasting  formula,  we  should  take  the 

equation  (y  —  y)  =  r  —  {x  —  x),  where  y  is  the  value  of 

the  predicted  yield-ratio  YJY^;y  is  the  mean  value  of  the 
actual  yield-ratios  ^JYf,  for  the  whole  period  under  inves- 
tigation; r  is  the  coefficient  of  correlation;  o-y  and  a-,  are, 
respectively,  the  variabilities  of  ^/Fs  and  C'/^^.;  x  is  the 
value  of  C'/^j.  for  the  current  year;  and  x  is  the  mean  value 
of  C'/Co  for  the  whole  period  under  investigation.  We 
know  from  the  theory  of  Chapter  II,  ''The  Mathematics 
of  Correlation,"  that  when  the  above  equation  is  used  as 
a  prediction  formula,  the  degree  of  accuracy  of  the  pre- 
diction is  measured  by  *S  =  (Ty\/l  —  r'-;  that  is  to  say,  the 
value  of  S  gives  the  root-mean-square  of  the  deviations 
of  the  actual  yield-ratios  from  the  predicted  yield-ratios. 

The  relative  accuracy  of  this  formula  as  compared 
with  the  official  formula  is  exhibited  by  the  calculations 
in  Table  14. 

In  Table  14,  r  measures  the  degree  of  correlation 
between  the  forecast  series  C'/c's  and  the  actual  yield- 
ratios  ^Iy^',  S,  as  we  have  indicated,  measures  the 
scatter  of  the  actual  yield-ratios  about  the  predicted 
yield-ratios,  when  the  forecast  is  made  by  means  of  the 

formula  iy  —  y)  =  r-^  {x  —  x);  S'  measures  the  scatter 
o-x 

of  the  actual  yield-ratios  about  the  predicted  yield- 
ratios,  when  the  forecasts  are  made  by  means  of  the 
official  formula. 

Table  14  shows,  by  the  magnitude  of  r,  that  the  May 
reports  have  no  value  and  that  the  reports  for  the  other 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports      99 


1^. 


JO 

o~ 


p 

K 

o 

CJ 

u 

z 

fa 

(M 

03 
->1 

O 

O 

H 

w 

ffi 

K 

E-i 

O 

U^ 

A^ 

b  I  b 


H 

P4 

"A 
O 

H 

1 

< 

Tt< 

a 

w 

w 

i-1 

z 
o 

<: 

<1 

H 

►J 

w 

K 

« 

O 

U 

0) 

JO 

a 

ft 

0) 

'-O 

CO 

00 
o 

o 

GO 

=0 

cc 

T-H 

o 

CO 

CO 

y-t. 

!- 

GO 

CO 

CO 
»M 

§8 

CO 

3 
M 

3 
< 

^ 

v: 

CO 
GO 

05 

CO 
CO 

8 

02 

v; 

CO 

CO 

o 

o 

(M 

CO 

1- 

-~ 

o 

CO 

CO 

<M 

CO 

OC 
(N 

CO 

3 

^^ 

o 

CM 

V; 

CO 

o 

CO 

d 

1— 1 

CO 
1— I 

'->. 

^ 

o 

o 

CO 
CO 

CD 

0! 

c 

V2 

o 

GO 

CO 

CO 

o 

05 

X 

o 

CO 

IM 

O 

00 

CO 

■^ 

05 

(N 

CO 

00 
CO 

CO 

^ 

CO 

00 

<2 

i-H 

o 

CO 

00 
CO 

CO 

CO 
■ 

CO 

CO 

00 

5^ 

CO 
O 

o 

t^ 

o 

00 

C    J! 

IB 

ft-x 

« ?; 

H 

■r. 

S3 
X 

o 

1 

'5c 

o 

o 

O 

£ 

< 

1 
O 

3 
O 
03 

100     Forecasting  the  Yield  and  the  Price  of  Cotton 

months  have  increasing  value  as  the  crop  approaches 
maturity.  The  comparative  values  of  S  and  S'  show 
that  in  every  month,  in  all  four  of  the  represent- 
ative states,  the  correlation  equation  gives  a  more 
accurate  forecast  than  the  official  formula.  Moreover, 
the  correlation  equation  as  a  forecasting  formula  does 
not  admit  of  the  anomalous  results  which  were  brought 
out  in  the  consideration  of  Table  13,  where  we  found 
that  in  a  number  of  cases  >S'  was  greater  than  ay,  which 
signifies  that  the  forecasts  are  worse  than  useless. 
When  the  correlation  equation  is  employed  as  a  fore- 
casting formula  it  is  impossible  for  *S  to  exceed  o-y, 
since  >S  =  c^v/l  —  r'. 

The  results  collected  in  the  two  Tables  in  this  section 
establish  two  of  the  theses  enunciated  at  the  beginning 
of  the  chapter: 

(1)  That  some  of  the  official  reports  referring  to  the 
representative  states  are  valuable  as  forecasts,  but  that 
others  are  worse  than  useless  in  the  sense  of  supplying 
erroneous  instruction  as  to  the  crop  outlook,  and 
thereby  suggesting  a  misdirecting  of  activity  on  the 
part  of  farmers,  dealers,  and  manufacturers; 

(2)  That  even  in  case  of  the  useful  forecasts  the 
official  method  does  not  extract  the  full  amount  of 
truth  contained  in  the  laboriously  collected  data. 

Forecasting  the   Yield  of  Cotton  from  the  Accumulated 
Effects  of  the  Weather  ^ 

Throughout  the  period  from  the  first  of  May  until  the 
end  of  September,  the  growth  of  the  cotton  plant  is 

'  Professor  J.  Warren  Smith,  of  the  Universitj^  of  Ohio,  and  Mr.  R.  H. 
Hooker,  of  London,  have  been  pioneers  in  dealing  with  special  phases 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports  101 

watched  with  anxious  sohcitude.  The  changes  of  the 
weather  may  convert  a  crop  that  is  flourishing  at  the 
first  of  June  into  a  comparative  failure  at  the  time  of 
harvest,  or  the  damage  of  excessive  heat  in  July  may  be 
off-set  by  a  beneficial  rainfall  in  August.  The  effects 
of  temperature  and  rainfall  upon  the  crop  vary  from 
state  to  state,  but  in  all  cases  the  effects  are  cumulative, 
and  the  probable  consequences  of  a  rain  or  drought  at 
any  point  in  the  growth  season  are  dependent  upon  the 
quantity  and  distribution  of  rainfall  and  temperature 
preceding  the  time  in  question.  The  principal  difficulty 
in  forecasting  the  yield  of  the  crop  from  the  changes  in 
the  weather  is  that  there  are  so  many  variables  in  the 
problem  and  all  of  the  variables  are  interrelated.  In  a 
particular  state  it  may  be  that  the  growing  plant  needs 
a  cool,  dry  July  and  a  rainy,  hot  August;  but  tempera- 
ture and  rainfall  may  be  so  interrelated  that,  on  the 
average,  in  July  when  the  weather  is  cool,  it  is  likewise 
rainy,  and  in  August  when  the  weather  is  hot,  it  is  also 
dry;  and  it  might  be  more  important  that  the  crop 
should  have  rain  in  August  than  that  July  should  be 
cool.  The  economist  who,  at  any  given  time  in  the 
growth  period,  seeks  to  forecast  the  yield  of  cotton  at 
harvest  must  be  able  to  measure  the  effects  upon  the 
crop  of  the  accumulated  variations  in  temperature  and 
rainfall  up  to  the  time  in  question. 

Before  passing  to  the  account  of  the  method  adopted 
to  measure  the  accumulated  effect  of  the  weather  upon 

of  the  topic  treated  in  this  section.  Professor  Smith  was  one  of  the 
first  to  see  the  economic  importance  of  forecasting  the  jdeld  of  the  crops 
from  the  weather,  and  Mr.  Hooker,  as  far  as  1  know,  led  the  way  in  the 
use  of  the  method  of  multiple  correlation  to  measure  the  joint  effect  of 
temperature  and  rainfall  upon  the  yield  of  the  crops. 


102    Forecasting  the  Yield  and  the  Price  of  Cotton 

the  growing  crop,  we  shall  consider  the  device  for  meet- 
ing the  difficulties  that  are  traceable  to  the  secular 
trend  and  cyclical  variations  in  the  yield  per  acre,  the 
rainfall,  and  the  temperature.  In  some  states  the  yield 
per  acre  of  cotton  throughout  the  period  under  investi- 
gation has  steadily  increased,  while  in  other  states  it 
has  steadily  decreased.  Moreover,  in  all  of  the  states 
the  yield  per  acre,  temperature,  and  rainfall  have,  dur- 
ing the  same  interval,  been  subjected  to  cyclical  in- 
fluences. In  order  to  measure  the  relation  between 
the  yield  per  acre  and  the  variations  in  the  weather, 
we  must  make  allowance  for  these  secular  and  cyclical 
changes,  and  to  do  this  we  have  profited  by  the  ex- 
perience of  the  Bureau  of  Statistics  of  the  Department 
of  Agriculture.  We  recall  that,  in  order  to  forecast 
from  the  condition  of  the  crop  in  any  month  the  prob- 
able yield  of  the  crop  at  the  end  of  the  year,  the  Bureau 
of  Statistics  does  not  work  directly  with  the  absolute 
values  of  the  condition  and  the  yield,  but  it  takes  the 
condition-ratio  and  the  yield-ratio,  the  forecasting 
formula  being  QCo  =  '^  IVb-  In  the  preceding  chapter 
we  showed  that  equally  good  results  would  be  obtained 
by  using  the  formula  QCs  =  ^  /Ys'}  that  is  to  say,  in- 
stead of  making  the  denominators  five  years  averages, 
to  employ  three  years  averages.  One  advantage  of  the 
latter  method  is  that  when  the  a^•ailable  data  are  few 
and  as  many  as  possible  must  be  utilized,  the  three 
years  method  gives  a  larger  number  of  cases  upon  which 
to  base  one's  computations. 

We  shall  use  this  three  years  method  in  correlating 
the  temperature-ratios  and  rainfall-ratios  of  the  several 
months  with  the  yield-ratios  of  cotton.    For  each  month 


Forecasting  the  Yield  oj  Cotton  from  Weather  Report     103 

the  series  to  be  correlated  will  be  ^/Ts,  ^/Fs;  ^IRz,  ^IYs- 
In  these  formulae  T  is  the  average  temperature  for  the 
given  month,  and  Ts  is  the  average  temperature  for 
the  same  month  during  the  preceding  three  years;  R  is 
the  amount  of  rainfall  for  the  given  month,  and  ^3  is 
the  average  amount  of  rainfall  for  the  same  month 
during  the  preceding  three  years ;  Y  is  the  yield  per  acre 
of  cotton  for  the  given  year,  and  F3  is  the  average  yield 
per  acre  of  cotton  for  the  three  years  preceding  the 
given  year.  In  Table  15  the  method  of  preparing  the 
data  for  computing  the  correlation  is  illustrated  by  the 
correlation,  in  Georgia,  between  the  June  temperature- 
ratio  and  the  yield-ratio  of  cotton. 

When  the  two  series  that  are  given  in  columns  VIII 
and  IX  of  Table  15  are  correlated,  it  is  found  that 
r  ^  .551,  and  this  value  of  r  gives  for  the  value  of  the 
scatter,  >S  =  0-^  \/F^-  =  13.89  s/H^-  =  11.59.  By 
referring  to  Table  13  we  find  that  the  official  method 
of  forecasting  from  the  condition  of  the  crop  gives  for 
the  month  of  June,  in  Georgia,  S'  =  15.20.  The  official 
forecast,  inasmuch  as  <Ty  =  13.89,  is  worse  than  useless, 
while  the  forecast  of  the  crop  from  the  June  tempera- 
ture has  a  decided  value.  We  also  know  from  Table  13 
that  the  official  method  of  forecasting  from  the  condi- 
tion of  the  crop  gives  for  the  month  of  May,  in  Georgia, 
a  value  of  *S'  =  17.28,  which  is  also  worse  than  useless. 
But  the  correlation  between  the  May  rainfall-ratio  in 
Georgia  and  the  yield-ratio  in  Georgia  gives  ?•  =  —  .410, 
and  S  =  12.07.  These  two  illustrations  show  that  the 
cotton  crop  in  Georgia  is  favorably  affected  by  a  dry 
May  and  a  warm  June.  How  would  it  be  possible  to 
utilize  the  knowledge  both  of  the  rainfall  in  May  and 


104     Forecasting  the  Yield  and  the  Price  of  Cotton 


o 

E- 

' 

H 

O 

t3  2         i>^ 

CO 

-f 

0 

M 

X 

•I* 

^ 

-f 

O 

X 

^ 

X 

o 

rt 

o 

35 

35 

CO 

M 

O 

^ 

M 

-r 

^ 

X 

35 

^ 

— 

-t> 

o 

-0 

t^ 

^ 

C^) 

rt 

c-q 

_H 

^ 

X 

I- 

1^ 

o 

tx> 

(M 

M 

35 

3-. 

a> 

~ 

33 

c^ 

;::! 

X 

o 

E 

35 

CO 

X 

o 

o 

o 

H 

,       ,       ,       ,       ,       , 1 

< 

rt 

3 

S 

^         <U  C3  2           icT 

o 

-t< 

-t^ 

-O 

03 

O 

X 

S 

CO 

■M 

CO 

-f 
t^ 

o 

X 

-r 

•V 

CO 

35 

lO 

o 

o 

o 

O 

O 

01 

o 

O 

o 

O 

s 

35 

S 

o 

35 

35 

o 

>^ 

^ 

"^ 

^ 

*"* 

'^ 

""* 

■"* 

'"' 

'~* 

^^ 

^^ 

H 

H 

» 

H 

9 

§  ft    UQ 

o 

00 

CO 

^ 

^ 

CO 

^ 

:C 

X 

CO 

■■O 

X 

O 

lO 

Csl 

X 

N 

^ 

C^ 

^ 

o 

M 

>o 

^ 

-f 

lO 

--0 

r» 

t^ 

■-C 

2 

■X 

:^ 

X 

35 

X 

X 

X 

X 

2 

2 

oi 

•< 

1 

£> 

P5 

iH  Ml  ca  a 

0 

o  S.<^  S 

E^ 

„     "^-^^^  S 

T-t 

CO 

05 

C<1 

CO 

o 

•* 

X 

'i' 

o 

X 

CO 

O 

m 

m 

-1< 

r^ 

t^ 

CO 

r-4 

< 
i 

k7        d  0)  -,ji 

lO 

Ttf 

C^ 

■o 

00 

C-l 

o 

o 

35 

IN 

to 

r- 

lO 

Tt< 

CO 

•* 

(35 

1^ 

>      9  S  g-o 

-^    0)    r  r'v 

H.9 

t 

Tt< 

■^< 

-V 

Tf 

■n 

't 

lO 

'I' 

lO 

lO 

lO 

■o 

■o 

1.0 

>o 

lO 

■n 

CO 

H 

H 

0)  "-<   O 

s 
» 

fto  CI  m^^ 

>     2«2'S.9t^ 

O 

'^ 

■o 

C<I 

C<l 

00 

CO 

05 

■M 

r- 

in 

X 

lO 

o 

lO 

o 

o 

-t< 

CO 

o 

CO 

00 

05 

O 

CO 

lO 

lO 

C) 

t^ 

00 

lO 

t^ 

!D 

to 

IQ 

o 

o 

•x> 

35 

35 

X 

t^ 

"V 

CO 

^ 

!N 

(» 

C^l 

s 

(N 

1^ 

U 

' 

1 



' 









H 

0)              S           -^     qj 

t^ 

o 

•-0 

to 

00 
00 

O 

00 

X 

t^ 

M 

I^ 

X 

O 
X 

t^ 

CO 

t^ 

<N 

X 

!N 

CO 

1^ 

t^ 

t^ 

t^ 

r^ 

00 

t^ 

t^ 

t~ 

r- 

p 

t^ 

t^ 

t^ 

t^ 

t~ 

t^ 

t^ 

t^ 

Z 

, 

o 

M  2'"' 

5 

1:3     ^"^^^i 

^ 

05 

r^ 

^ 

•r: 

^ 

_, 

rf 

t^ 

O 

CO 

lO 

35 

uO 

CO 

—J 

CO 

1^ 

CO 

t:~ 

h3 

o 

05 

c<i 

to 

X 

_; 

lO 

CO 

(N 

^ 

o 

CO 

C^ 

^ 

!N 

^ 

^ 

^ 

M 

Pi 

o 

CO 

C^l 

CO 

CO 

^ 

CO 

CO 

TO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

CO 

d  ?^  «  o 

C^l 

(M 

IN 

(N 

c^i 

a 

C^ 

(M 

!N 

c5 

N 

(N 

c^ 

O) 

c5 

N 

c^ 

(N 

?N 

C^l 

O 

H.S 

< 

c2i 

.      §5 

o 

tf 

§£ 

■-1 

CT> 

— 1 

■35 

1^ 

CO 

'-' 

•M 

■■D 

-n 

lO 

•'f 

T)< 

t~ 

X 

o 

''? 

"^ 

^ 

35 

Tt" 

rfi 

N 

o 

s     .a     ^ 

00 

^ 

t^ 

I- 

r~ 

o 

o 

o 

lO 

t^ 

05 

Tf 

t^ 

00 

t- 

CO 

r^ 

X 

o 

s 

>o 

CD 

Ol 

H 

i^ 

t^ 

t^ 

■X 

CO 

X 

t^ 

t^ 

t^ 

t^ 

t^ 

t^ 

t^ 

I^ 

t^ 

r- 

t^ 

X 

t^ 

t^ 

X 

o 

1 

lO 

T-H 

w 

e3 

CO 

■^ 

lO 

•s 

t^ 

00 

o> 

o 
o 

-^ 

<N 

CO 

■S< 

in 

■■D 

t^ 

X 

05 

o 

;z; 

CT 

CO 

•* 

1-3 

a 

<i 

1 

Eh 



1 

1     1     i 

1 

' 

' 



! L — 1 — 1 — ! — ! 

Forecasting  the  Yield  of  Cotton  from  Weather  Reports  105 

the  temperature  in  June  to  forecast,  at  the  end  of  June, 
the  probable  yield  of  the  crop?  This  question  brings  us 
to  a  consideration  of  the  method  of  multiple  correlation. 

In  Chapter  II,  "The  Mathematics  of  Correlation," 
we  developed  in  considerable  detail  the  theory  of  the 
correlation  between  two  variables.  The  essential  steps 
are: 

(1)  The  assumption  ^  that  the  two  variables  are 
related  in  a  linear  way  bj^  the  equation  y  =  mx  +  6; 

(2)  The  calculation  of  the  coefficient  of  correlation 
r;  and 

(3)  The  determination  of  the  accuracy  of  the  equa- 
tion y  =  mx  +  6  as  a  forecasting  formula  by  calculating 
the  scatter  S  =  o'^s/l  —  r-.  In  the  theory  of  multiple 
correlation  the  essential  steps  run  parallel  to  those  in 
the  theory  of  the  correlation  of  two  variables.  In  case 
of  three  variables  the  three  steps  are : 

(1)  The  assumption  that  the  equation  connecting 
the  three  variables, 

OJ  It       ^  '    {1  0       I        CtiXi        I        Cl/fy^O  1 

(2)  The  determination  of  the  degree  of  association 
between  the  variable  Xo  and  the  other  two  variables 
Xi,  Xi  by  calculating  the  coefficient  of  multiple  correla- 
tion R; 

(3)  The  determination  of  the  accuracy  of  the  equa- 
tion Xo  =  cto  +  «i^*i  +  a2'^'2  as  a  forecasting  formula  by 
calculating  the  scatter,  *S"  =  a^  \/l  —  R-. 

We  found,  in  the  theory  of  the  correlation  of  two  vari- 

*  There  are  methods  for  testing  the  legitimacy  of  the  assumption 
which  should,  of  course,  be  ai)i)li(Ml. 


106     Forecasting  the  Yield  and  the  Price  of  Cotton 

ables,  that  the  forecasting  formula,  namely,  y  =  mx  +  ^, 
may  be  put  into  the  form  {y  —  y)  =  m{x  —  x).  In  a 
similar  manner,  when  three  variables  Xo,  Xi,  Xi,  are 
correlated,  the  forecasting  formula  may  be  put  into  the 
form 

{xq  —  Xo)  =  ai{xi  —  Xi)  +  a2{x2  —  Xi). 

Our  statistical  problem  will  be  solved  if  we  can  de- 
termine from  the  statistical  data  the  following  three 
items : 

(1)  The  values  of  the  coefficients  of  {xi  —  X\)  and 
{x2  —  X'l)  in  the  forecasting  formula. 
These  values  are 

_   ^01  ^02'"l2  ^.  _   ^n2  ^'(11^12  <^n 


1  —  r\^    (Ti  1  —  ri2    0-2 

Here  r^^  is  the  coefficient  of  correlation  between  the 
variables  rCp,  x^\  r^^  is  the  coefficient  of  correlation  be- 
tween Xq  and  Xo;  ri2  the  correlation  between  x^  and  Xi) 
(7o  is  the  standard  deviation  of  the  x^'s;  o-^  is  the  stand- 
ard deviation  of  the  Xi's;  and  (Ji  the  standard  deviation 
of  the  X2's.  When  these  values  of  a\,  a^  are  substituted 
in  the  forecasting  formula  (Xq  —  x^)  =  ai{xi  —  Xi)  + 
a2(x2  —  X2),  the  most  probable  value  of  Xq  may  be 
calculated  from  the  known  values  of  Xi,  x-z. 

(2)  The  value  of  the  coefficient  of  multiple  correla- 
tion, R.    In  case  of  three  variables  Xq,  Xi,  X2, 

p2    _    ^01     I      ^02  ■^^01^02^12 

l~rl 

(3)  The  value  of  the  scatter,  S"  =  cr^s/l  —  R-, 
which  measures  the  root-mean-square  of  the  devia- 
tions of  the  observed  values  of  Xo  from  the  most  prob- 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports  107 

able  values  of  x^^  when  the  most  probable  values  are 
predicted  from  the  forecasting  formula  {xq  —  Xo)  = 
ai{xi  —  Xi)  +  a^ixi  —  x^). 

We  may  proceed  at  once  to  illustrate  the  method  by 
showing  how  the  yield-ratio  of  cotton,  in  Georgia, 
may  be  predicted  from  the  rainfall-ratio  for  May  and 
the  temperature-ratio  for  June.  Let  the  yield-ratio 
be  Xo,  the  rainfall-ratio  for  May  be  X\,  and  the  tempera- 
ture-ratio for  June  be  x^.     The  forecasting  formula  is 

(.To  —  Xo)  =  ai{xi  —  Xi)  +  a-iix-i  —  x^). 

From  the  statistical  data  we  find  that 

(1)  The  mean  values  of  Xo,  Xi,  .To,  are,  respectively, 

X,  =  104.05;  xi  =  107.04;  x.  =  100.34; 

(2)  The  standard  deviations  of  Xq,  Xi,  x^  are,  re- 
spectively, 

0-0  =  13.890;  0-1  =  65.528;  a.  =  3.142; 

(3)  The  coefficient  of  correlation  between  Xo  and  Xi 
=  roi  =  —  .410;  between  Xo  and  X2  =  r,^  =  .551;  be- 
tween Xi  and  Xg  =  r^^  =  —  .427. 

Since   ai  ^^01-^12^^    ^^^    ^^  =  ''"^  ~  Y'"  ">   the 
1  -  r?2     0-1  1  —  rl,     0-, 

substitution  of  the  above  numerical  values  for  the 
algebraic  symbols  gives  cti  =  —  .045,  and  a2  =  2.033. 
After  the  proper  substitutions  have  been  made  and  the 
equation  simplified,  the  forecasting  formula  becomes 
Xo  =  —  95.12  -  .045x1  +  2.033x2.  * 
Since  R'  =  ^01  +  ^02  -  2^ir,„ri,^  ^  ^         ^^^  ^^^  coefficient 

1-^12 
of  correlation  between  Xo  and  the  two  variables  Xi,  X2, 


108     Forecasting  the  Yield  and  the  Price  of  Cotton 

the  value  R-  =  .340933,  or  R  =  .584.  Furthermore, 
since  S"  =  o-q  v/l  —  R-,  we  get  as  the  numerical 
measure  of  the  accuracy  of  the  forecasting  formula, 
iS"=  11.28. 

We  have  seen  that,  by  means  of  the  forecast  from  the 
weather,  we  get  a  formula  for  reducing  the  variability 
of  o-Q  even  in  May,  while  the  Government  report  for 
May  we  have  found  to  be  erroneous  and  misleading. 
Furthermore,  by  means  of  the  additional  information 
given  by  the  weather  reports  for  June,  we  have  been 
able  still  further  to  reduce  the  variability  of  a^.  The 
value  of  S"  =  o-q  s/l  —  R-  for  June  is  found  to  be 
11.28.  We  may  at  this  point  take  stock  of  our  gains. 
From  Table  13  we  know  that,  according  to  the  official 
method,  the  accuracy  of  the  forecasts  are,  for  May, 
S'=  17.28;  for  June,  S' =  15.20;  for  July,  S' =  12.17; 
for  August,  ^'=11.92;  and  for  September,  S'  ^  11.08. 
But  by  means  of  the  method  of  forecasting  from  the 
data  of  the  weather,  we  get,  for  May,  S"  =  12.67;  and 
for  June,  >S"  =  11.28,  where,  in  the  June  forecast,  we 
use  the  rainfall-ratio  for  May  and  the  temperature- 
ratio  for  June.  Not  only  are  the  forecasts  from  the 
weather  for  these  two  months  better  than  the  fore- 
casts by  the  official  method  from  the  condition  of  the 
crop,  but  the  value  of  S"  for  May  is  about  as  good  as 
the  value  of  S>'  two  months  later,  at  the  end  of  July; 
and  the  value  of  S"  for  June  is  about  as  good  as  the 
value  of  S'  two  or  three  months  later  at  the  end,  re- 
spectively, of  August  and  September. 

But  our  method  admits  of  still  further  usefulness. 
From  the  accompanying  Table  16,  we  see  that  the 
fruitfulness  of  the   cotton  crop  is  affected  not  only 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports  109 

by  the  weather  of  May  and  June  but  also  by  that 
of  July  and  of  August.  The  coefficients  for  both  tem- 
perature and  rainfall  in  August  have  significant  values. 
If  at  the  end  of  August  we  should  wish  to  forecast  the 
yield  of  cotton  from  accumulated  effects  of  past  weather 
—  for  example,  of  the  May  rainfall,  the  June  tempera- 
ture, and  the  August  temperature  —  we  have  only  to 
extend  the  principles  of  multiple  correlation  to  cover 
four  variables.  The  steps  in  the  development  are  again 
three  in  number  and  they  run  parallel  with  the  three 
steps  that  we  have  already  traversed  in  describing  the 
correlation  between  two  variables  and  the  correlation 
between  three  variables. 

TABLE  16.  —  Georgia.    Correlation  Between  the  Yield-Ratio 
OF  Cotton  and  the  Temperature-Ratio  and  Rainfall-Ratio 


Values  of  the  CoeflScient  of  Correlation 

May 

June 

July 

August 

September 

Temperature-Ratio 

—  .097 

.551 

—  .032 

—  .499 

.082 

Rainfall-Ratio 

—  .410 

—  .411 

—  .254 

.426 

—  .188 

The  three  steps  are : 

(1)  The  assumption  that  the  equation  connecting  the 
four  variables  x^,  X\,  x^,  Xz  is  x^  =  (Zq  +  c^i^i  +  ^2X2  + 
azXz'  It  is  not  difficult  to  prove  that  this  forecasting 
equation  may  be  put  into  the  form 

(xo  —  Xo)  =  ax{xi  —  Xi)  +  ^2(^2  —  ^2)  +  az{xz  —  Xz). 

By  the  Method  of  Least  Squares  the  values  of  the  co- 
efficients in  the  forecasting  equation  are  ascertained  to  be 

nn(l  —  rjs)  +  ro2(rur2s  —  ^12)  +  rozir^r^z  -  ru)  (To^ 
(1  —  r^g)  +  ruirizroz  -  ^-12)  +  riziri2r23  -  riz)  <ri' 


fli  = 


no    Forecasting  the  Yield  and  the  Price  of  Cotton 


ro2  (1  - 

~  Ms)  ~l~  ^03(^12^13  " 

~  '"23)  +   '"0t(7"l3^23    ~ 

-  rn)  (To 

Q/>  — 

(1- 
ro3  (1  - 

-  Az)  +  ^23Cn2ri3  - 

-  4)  +  r^Sxnru  - 

"  ^23)  ~r   ^12(Ti3^23  ~ 

-  ^23)  +  roi(r,2r23  - 

-  ^12)  0-2  ' 

-  rn)  (Tu 

«3    — 

(1- 

-r\o)  +  r^zirnrn  - 

~  ''23)  +  ^3  (^12^23  ~ 

-  rn)  (Ts 

(2)  The  determination  of  the  degree  of  association 
between  the  variable  Xq  and  the  other  three  variables 
by  calculating  the  coefficient  of  multiple  correlation  R. 
In  the  case  of  four  variables  the  value  of  R  is  given 
by  the  following  equation: 

R"  = where 

1  -  7^2  -  r^3  -  7^3  +  2r  1 2ri  gr  2  3' 

I   ^"         I        y»^         I       ry^     /v*^    rv*^      /v*       "Y^"     'J'*'    'X*         1 

V  01    1^  '  02    l^  '  03        '01' 23        '02' 13        '03' 12/ 

M=  <         — 2(roiro2r"i2+roiro3ri3+ro2ro3r23) 

l^  ~r  2(roi^02^is'^2  3~r^*01^0  3'^12^2  3~r  ^02^0  3^12^'l3)  J 

(3)  The  determination  of  the  accuracy  of  {x.  —  Xq)  = 
ai{xi  —  Xi)  +  a2(x2  —  ^^2)  +  03(3:3  —  X3)  as  a  forecasting 
formula  by  calculating  the  scatter,  S"  =  <Jq  \/i  —  RK 

To  illustrate  the  use  of  this  more  complex  forecasting 
formula  we  shall  go  through  the  work  of  ascertaining 
how  the  yield-ratio  x^  may  be  predicted  from  the 
knowledge  of  three  variables,  Xx  =  May  rainfall-ratio, 
X2  =  June  temperature-ratio,  Xz  =  August  temperature- 
ratio. 

From  the  statistical  data  we  find  that 

(1)  The  mean  values  of  Xo,  a;i,  X2,  x^  are,  respectively, 

xo  =  104.05;  rci  =  107.04;  X2  =  100.34;  Xz  =  100.15; 

(2)  The  standard  deviations  of  a^^,  x^^  x.2,  x^  are,  re- 
spectively, 

0-0  =  13.890;  (7i  =  65.528;  0-2  =  3.142;  0-3  =  1.761; 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports  111 
(3)  The  coefficients  of  correlation  are 

roi  =  -  .410;  ro2  =  .551;  r,,  =  -  .499;  r^.  =  -  .427; 
ri3=  .014;  r23=  -  .126. 

WTien  these  numerical  values  are  substituted  for  the 
algebraic  symbols  in  the  above  formulae,  we  obtain 
for  the  forecasting  formula,  Xo  =  286.84  —  .OSOa^i  + 
1.743x'2  —  3.518X3;  for  the  coefficient  of  multiple  corre- 
lation, R  =  .732;  for  the  scatter,  S"  =  9.46. 

In  Figure  9  the  continuous  line  represents  the  values 
of  the  yield-ratios  for  the  several  years  as  the  yield- 
ratios  are  computed  from  the  actual  statistics  by  means 
of  the  formula  Y/y^-  the  dashed  line  represents  the 
yield-ratios  as  they  are  computed  from  the  rainfall- 
ratio  of  IMay,  the  temperature-ratio  of  June,  and  the 
temperature-ratio  of  August  by  means  of  the  forecasting 
formula 

Xo  =  286.84  -  .050x1+  1.743x2  -  3.518x3; 

The  root-mean-square  deviation  of  the  actual  ratios 
from  the  predicted  ratios  is  S"  =  9.46. 

Figure  10  illustrates  the  degree  of  precision  in  the 
forecast  at  the  end  of  August,  by  means  of  the  official 
method,  of  the  yield  of  cotton  in  Georgia.  The  contin- 
uous line  represents  the  value  of  the  actual  yield-ratios, 
computed  by  the  formula  Y/y^;  and  the  dashed  line 
represents  the  theoretical  yield-ratios  as  they  are  pre- 
dicted by  the  formula  ^/Cy  The  root-mean-square 
deviation  of  the  actual  ratios  from  the  predicted  ratios 
is  S'=  11.92. 

If  now  we  compare  the  value  of  S"  for  August  with 
the  value   of   *S'  for    September,   that  is,  aS' =  11.08, 


112       Forecasting  the  Yield  and  the  Price  of  Cotton 


-o/^^^-p/3//' p3/o/p3^(/  ^u^  puo  o//oj'pf3/X^  ^on^Y 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports    113 


-  'i 


o/^-^-pp// ps^o'psjc/ 3u/ piJo  o/^o^-p/a//'  /on^3\f 


114     Forecasting  the  Yield  and  the  Price  of  Cotton 

we  see  that  from  the  weather  data  we  can  obtain,  by 
means  of  mathematical  methods,  a  better  forecast 
at  the  end  of  August  than  the  official  method  enables 
us  to  obtain  from  the  condition  of  the  crop  at  the  end 
of  September.  A  clearer  view  of  the  value  of  fore- 
casting the  cotton  yield  from  the  weather  reports,  by 
means  of  the  methods  that  we  have  described,  may  be 
had  from  the  following  Table  17. 

TABLE  17.  — •  Relative  Accuracy  of  the  Forecasts  of  the  Yield 
Per  Acre  of  Cotton  (1)  from  the  Condition  of  the  Crop,  by 
THE  Official  Method,  and  (2)  from  the  Weather  Reports, 
BY  the  Method  of  Correlation.    Georgia 


Months 

May 

June 

July 

August 

September 

Error 

of  the 

Forecasts 

S' 
S" 

17.28 

15.20 

12.17 

11.92 

11.08 

12.67 

11.28 

9.94 

9.46 

9.46 

In  computing  >S"  we  used  for  May,  the  rainfall-ratio; 
for  June,  the  May  rainfall-ratio  and  the  June  tempera- 
ture-ratio; for  July,  the  May  rainfall-ratio,  the  June 
temperature-ratio,  and  the  July  rainfall-ratio;  for 
August  and  September,  the  May  rainfall-ratio,  the 
June  ten  perature  ratio,  and  the  August  temperature- 
ratio.  From  Table  17  it  is  seen  that  not  only  is  S" 
less  than  S^  for  every  month,  but  S"  for  May  is  about 
as  good  as  S'  two  months  later,  at  the  end  of  July; 
the  S"  for  June  is  about  as  good  as  the  S'  two  months 
later,  at  the  end  of  August;  the  *S"  for  July  is  better 
than  the  S'  two  months  later,  at  the  end  of  September; 
and  the  S"  at  the  end  of  August  is  better  than  the  S' 
at  the  end  of  September. 


Forecasting  the  Yield  of  Cotton  fro7n  Weather  Reports  115 

As  far  as  the  state  of  Georgia  is  concerned  we  have 
proved  our  thesis:  That  it  is  possible,  by  means  of  the 
weather  reports  and  mathematical  methods,  to  fore- 
cast the  yield  per  acre  of  cotton  with  a  greater  degree 
of  precision  than  the  reports  of  the  official  Bureau  with 
its  vast  organization  for  the  collection  and  reduction 
of  data  referring  to  the  condition  of  the  growing  crop. 

The  Results  Compared  for  the  Representative  States 

The  thesis  that  has  just  been  proved  for  the  state  of 
Georgia  we  shall  now  test  for  the  representative  states 
Texas,  Georgia,  Alabama,  and  South  Carolina,  which 
together,  in  1914,  produced  65  per  cent  of  the  total 
crop  of  the  United  States.  In  Table  18  are  exhibited 
the  correlations  between  the  yield-ratios  of  cotton  and 
the  temperature-ratios  and  rainfall-ratios  of  the  re- 
presentative states.^     If  we  refer  to  the  earlier  part 

1  The  statistics  of  the  condition  of  the  crop  and  of  the  yield  per  acre 
of  cotton  were  kindly  supplied  to  me  bj'  Mr.  Leon  M.  Estabrook  and 
Mr.  George  K.  Holmes  of  the  U.  S.  Department  of  Agriculture.  The 
figures  are  reproduced  in  Tables  23,  24,  25,  26  of  the  Appendix  to  this 
chapter.  The  weather  data  for  Georgia,  Alabama,  and  South  Carolina 
were  taken  from  the  pubUcation  of  the  U.  S.  Weather  Bureau  Cli- 
matoloc/ical  Data  for  1915,  and  refer,  in  each  case,  to  the  mean  tempera- 
ture and  mean  rainfall  for  the  entire  state.  The  coefficients  of  correla- 
tion between  the  yield-ratios  and  the  weather-ratios  were  based  upon 
20  ratios  in  case  of  Georgia  (1895-1914);  17  ratios  in  case  of  Alabama 
(1898-1915);  21  ratios  in  case  of  South  Carolina,  and  21  ratios  in  case 
of  Texas  (1894-1914).  Because  of  the  great  size  of  Texas  and  the  con- 
centration of  tJie  cotton  production  in  the  Eastern  and  Central  parts 
of  the  state,  the  mean  temperature  and  mean  rainfall  were  computed 
for  those  two  sections  from  the  records  for  the  individual  stations  that 
are  given  in  the  Annual  Reports  of  the  Chief  of  the  U.  S.  Weather 
Bureau.  The  thirty-one  stations  that  were  selected  for  the  rainfall 
record  were:  Abilene,  Albanj-,  Austin,  Brenham,  Brownwood,  Clayton- 
ville,  Coleman,  College  Station,  Corsicana,  Dallas,  Fairland,  Fort 
Worth,   Fredericksburg,   Gainesville,   Graham,   Greenville,   Huntsville, 


116     Forecasting  the  Yield  and  the  Price  of  Cotton 


^  CO 

Is 


O 

5 

H 

h 

?^ 

o 

W 

r« 

o 

W 

5 

W 

Q 

J 

?: 

W 

t-H 

>^ 

O 

H 

bi 

-< 

« 

>-] 

15 

>-! 

w 

<! 

bi 

b 

^ 

S? 

H 

< 

tf 

Z 

o 

»— I 

H 

-^ 

hJ 

w 

K 

« 

O 

U 

w 

a 

H 

'rt 

00 

GO 

(M 

I— 

1^ 

GO 

-<* 

to 

d 

0 

iM 

9 

'3 

,■ 

,• 

,■ 

J3 

a 

oj 

ft 

3 

ttj 

CO 

(M 

no 

»o 

CO 

cS 

CO 

GO 

05 

CO 

a; 

a 

a 

0 

0 

0 

H 

"3 

r^ 

CO 

!M 

0 

Oi 

(M 

(M 

t^ 

d 

0 

^ 

<M 

co_ 

'3 

^ 

« 

3 

3 

£ 

<lj 

3 

GO 

a> 

t^ 

T-H 

03 

(N 

OS 

t^ 

00 

.2 

1 

CO 

'^ 

(M 

CO 

<u 

cS 

£ 

H 

tH 

o 

o 

"3 

r^ 

-* 

CO 

0 

^-1 

10 

"O 

10 

*© 

a 
'3 

0 

(M 

IM 

'~| 

a 

tf 

1 

1 

■S 

_>. 

"3 

ui 

s 

5 

>* 

(N 

iJO 

00 

o 

C3 

t^ 

CO 

CO 

05 

(P 

Tf 

0 

0 

j3 

ft 

• 

■ 

C 

1 

1 

"o 

H 

_3 

3 

10 

,—1 

lO 

^H 

"3 

05 

1-H 

QC 

CO 

> 

(2 
3 

■3 

■*! 

CO 

00 

0) 

^^ 

u 

3 

t^ 

^H 

t^ 

05 

c3 

>* 

10 

Tt* 

0 

QJ 

0 

iC 

(M 

'^ 

8 

1 

(D 

H 

■3 

CO 

0 

00 

10 

»o 

»-H 

CO 

GO 

's 

0 

Tt< 

10 

'^ 

■3 

rt 

1 

1 

1 

>. 

' 

' 

' 

ca 

S 

^ 

3 

CO 

r^ 

»o 

0 

oS 

<o 

C5 

00 

CO 

0 

0 

r— I 

1-H 

a 

B 

1 

<D 

H 

<p 

tc 

.3 

c3 
S 

o3 

sl 

03 
a; 

'5c 

0 

0 

0   g 

& 

< 

0 

rt 

F(yrecasting  the  Yield  of  Cotton  from  Weather  Reports   117 

of  this  chapter  we  shall  see  that  the  test  of  the  accuracy 
of  the  official  crop  reports  rests  upon  the  yield-ratios 
and  condition-ratios  for  the  period  1894-1914.  In 
order  that  the  accuracy  of  the  forecasts  from  the  weather 
may  be  more  fairly  compared  with  the  accuracy  of 
the  forecasts  from  the  condition  of  the  crop,  the  period 
of  the  weather  observations  has  been  taken,  in  case 
of  each  state,  as  near  as  possible  to  the  period  1894- 
1914.  The  Texas  ratios  are  from  1894  to  1914;  those  of 
Georgia  from  1895  to  1914;  of  South  Carolina,  1894 
to  1914.  Because  of  the  hmited  meteorological  record 
in  Alabama  the  longest  series  of  ratios  that  could  be 
obtained  runs  from  1898  to  1915. 

The  raw  data  are  given  in  Tables  27,  28,  29,  30  of  the 
Appendix  to  this  chapter,  and  in  computing  the  co- 
efficients of  correlation  all  of  the  data  in  the  Tables 
have  been  used  exactly  as  they  are  recorded.^  It  would 
have  been  possible,  on  several  occasions,  to  increase 
the  coefficients  by  omitting  one  or  two  rainfall-ratios 
which,  in  consequence  of  torrential  storms,  presented 
unduly  large  values;  but  no  such  liberty  has  been  taken 
with  the  crude  material,  although  for  purposes  of  fore- 
casting the  yield  in  normal  times  such  a  procedure 
might  have  been  justifiable. 

Lampasas,  Longview,  Menardville,  Nacogdoches,  Palestine,  Panter, 
Paris,  San  Angelo,  Sulphur  Springs,  Taylor,  Temple,  Tyler,  Waco, 
Weatherford.  The  seventeen  stations  the  records  of  which  were  used 
in  compiling  the  mean  temperature  were  Abilene,  Brenham,  Brownwood, 
Corsicana,  Dallas,  Fort  Worth,  Greenville,  HuntsviUe,  Lampasas, 
Longview,  Nacogdoches,  Palestine,  Paris,  Taylor,  Temple,  Waco, 
Weatherford. 

The  weather  data  for  all  four  states  are  given  in  Tables  27,  28,  29, 
30,  of  the  Appendix  to  this  chapter. 

'  The  omission  of  the  August  rainfall  for  1914,  in  Texas,  is  recorded 
in  the  Notes  to  Table  19. 


118     Forecasting  the  Yield  and  the  Price  of  Cotton 

A  summary  view  of  the  relative  accuracy  of  the 
forecast  of  the  yield  from  the  condition  of  the  crop  and 
the  forecast  from  the  accumulated  rainfall  and  tempera- 
ture is  given  in  Table  19.  In  considering  the  following 
comments  on  Table  19  we  shall  bear  in  mind  that  S' 
measures  the  accuracy  of  the  forecasts  from  the  con- 
dition of  the  crop  by  the  official  method,  and  S",  the 
accuracy  of  the  forecast  from  the  weather  by  the  method 
of  correlation.  The  more  accurate  the  forecasts,  the 
smaller  are  the  respective  values  of  S'  and  S". 

(1)  There  are  four  representative  states  —  Texas, 
Georgia,  Alabama,  and  South  Carolina,  and  there  are 
five  monthly  reports  on  the  condition  of  the  growing 
crop,  the  reports  describing  the  condition  of  the  crop 
at  the  end  of  May,  June,  July,  August,  and  September. 
There  are,  therefore,  twenty  cases  in  which  the  accuracy 
of  the  two  methods  may  be  compared.  Table  19  shows 
that  in  17  out  of  20  cases  the  forecast  from  the  weather 
by  the  method  of  correlation  is  more  accurate  than  the 
forecast  by  the  official  method  from  the  condition  of 
the  growing  crop. 

(2)  For  all  of  the  representative  states  the  forecasts 
by  the  official  method  from  the  May  condition  of  the 
crop  are  worse  than  useless  because  the  values  of  S' 
are  larger  than  the  corresponding  values  of  a,j.  On  the 
contrary,  the  forecasts  from  the  May  weather  by  the 
method  of  correlation  have  a  real  value.  ^  The  forecasts 
from  the  weather  for  Georgia  and  South  Carolina  are, 
at  the  end  of  May,  better  ■  than  the  official  forecasts 
for  June,  and  nearly  as  good  as  the  official  forecasts 
at  the  end  of  July;  and  the  forecast  for  Alabama  at  the 

'  The  last  of  the  Notes  on  Table  19  should  be  consulted. 


Forecasting  the  Yield  of  Cotton  frovi  Weather  Reports   119 

end  of  May  is  nearly  as  good  as  the  official  forecast 
at  the  end  of  September. 

TABLE  19.  —  Relative  Accuracy  of  the  Forecasts  of  the  Yield 
Per  Acre  of  Cotton  (1)  from  the  Condition  of  the  Crop,  by 
yHE  Official  Method,  and  (2)  from  the  Weather  Reports, 
BY  the  Method  of  Correlation 


Repre- 
sentative 
States 

Standard 
Deviation 
of  the 
Yield- 
Ratios 

Error  of  the  Forecast  from  the  Condition  of  the  Crop,  .S" 
Error  of  the  Forecast  from  the  Weather,  S" 

May 

June 

July 

August 

September 

S' 

S" 

.S" 

S" 

S' 

S" 

6" 

S" 

5' 

S" 

Texas 

24.64 

26.38 

25.31 

22.11 

2b.  lb 

19.23 

22,58 

17.86 

16.80 

13.77 

16.80 

Georgia 

13.89 

17.28 

12.67 

15.20 

11.28 

12.17 

9.94 

11.92 

9.46 

11.08 

9.46 

Alabama 

13.37 
18 .  Go 

17.59 

10.40 

13.58 

9.70 

12.24 

9.52 

11.66 

9.19 

10.21 

9.19 

South 
Carolina 

21.90 

17.42 

19.06 

16.10 

17.02 

16.10 

19.03 

14.98 

15.28 

14.98 

In  computing  >S",  the  formula  .S"  =  »  /     •! -^^  V     lYh  I  C:J     t   was  used  for  everv 

\        I ^ J 

.state  and  every  month. 

In  obtaining  S"  the  formulas  .S"  =  a"„  \, '  1  —  r'-,  S"  =  cr^  v'^  ^  ~  ^'  ^'ere  used  accord- 
ing as  one  or  more  independent  variables  were  employed.  The  combinations  of  vari- 
ables were: 

In  case  of  Texas:  For  May,  temperature-ratio;  for  June,  rainfall-ratio;  for  July, 
temperature  ratio;  for  August  and  September,  July  temperature-ratio,  August  tem- 
perature ratio,  and  August  rainfall  ratio.  In  computing  the  correlation  between  the 
yield-ratio  and  the  rainfall-ratio  for  August,  the  rainfall  data  for  1914  were  not  u.sed. 

In  case  of  Georgia:  For  May,  rainfall-ratio;  for  June,  May  rainfall  ratio  and  June 
temperature-ratio;  for  July,  Slay  rainfall-ratio,  June  temperature-ratio,  and  Jul.v 
rainfall  ratio;  for  August  and  September,  May  rainfall-ratio,  .lune  temperature-ratio, 
August  temperature-ratio. 

In  case  of  Alabama:  For  May,  rainfall-ratio;  for  June,  May  rainfall-ratio,  Jime 
rainfall-ratio;  for  July,  May  rainfall-ratio,  June  rainfall-ratio,  July  rainfall-ratio;  for 
August  and  September,  May  rainfall-ratio,  June  rainfall-ratio,  Augu.st  temperature- 
ratio. 

In  case  of  South  Carolina:  For  May,  rainfall-ratio;  for  June  and  July,  May  rainfall- 
ratio,  June  rainfall-ratio  and  June  temperature-ratio;  for  August  and  September,  May 
rainfall-ratio,  June  temperature-ratio,  August  temperature-ratio. 

The  ratios  that  were  correlated  were  obtained  from  the  official  statistics  by  means 

of  the  formulas      hp         jB  y     where  the  symbols  refer,  respectively,  to  the  tem- 

perature, rainfall,  and  yield  per  acre  of  cotton. 

The  values  of  cr„  are  the  standard  deviations  of  the  yield-ratios  when  the  five  years 
progressive  means  are  used.  This  will  explain  the  rather  anomalous  result  that  S", 
in  case  of  Texas,  is  for  May  and  June  larger  than  c,..  When  the  three  years  means 
are  the  basis  of  the  yield-ratios,  the  standard  deviation  is  25.65. 

(3)  For  three  out  of  the  four  states  the  prediction 
from  the  June  condition  of  the  crop  by  the  official 
method  is  woi'se  than  useless  since  the  \'alues  of  S' 
are  greater  than  the  corresponding  values  of  a,j.     But 


120     Forecasting  the  Yield  and  the  Price  of  Cotton 

the  forecasts  from  the  weather  are  in  all  three  cases 
of  decided  value,  being  in  all  three  cases  better  than 
the  official  forecasts  for  the  following  month. 

(4)  For  all  of  the  states  except  Texas  the  forecast 
from  the  weather  gives,  for  each  month,  a  more  accu- 
rate prediction  than  can  be  obtained  by  the  official 
method  from  the  condition  of  the  crop  one  month 
later.  That  is  to  say,  for  all  of  the  states  except  Texas, 
>S"  for  May  is  smaller  than  S'  for  June;  S"  for  June  is 
smaller  than  S'  for  July;  *S"  for  July  is  smaller  than  S' 
for  August;  and  S"  for  August  is  smaller  than  S'  for 
September. 

(5)  For  all  of  the  states  except  Texas  the  forecasts 
from  the  weather  give  for  May,  June,  and  July  about 
as  good  predictions  as  can  be  obtained  by  the  official 
method  from  the  condition  of  the  crop  two  months  later. 
(In  six  out  of  the  nine  possible  cases  S''  is  less  than  S' 
two  months  later.) 

Considering  the  character  of  these  findings  it  is 
clear  that,  as  far  as  concerns  the  representative  states 
which  produce  sixty-five  per  cent  of  the  total  cotton 
crop  of  the  United  States,  we  may  conclude,  in  terms  of 
our  thesis:  "Notwithstanding  the  vast  ofl&cial  organi- 
zation for  collecting  and  reducing  data  bearing  upon  the 
condition  of  the  growing  crop,  it  is  possible,  by  means 
of  mathematical  methods,  to  make  more  accurate 
forecasts  than  the  official  reports,  in  the  matter  of  the 
prospective  yield  per  acre  of  cotton,  simply  from  the 
data  supplied  by  the  Weather  Bureau  as  to  the  current 
records  of  rainfall  and  temperature  in  the  respective 
cotton  states." 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports   121 

Three  Possible  Objections 

The  substance  of  this  section,  whicli  is  technical  in 
character,  is  intended  to  meet  three  quite  natural  ob- 
jections: 

(1)  In  the  preceding  chapter  and  in  the  early  part 
of  the  present  chapter,  the  defects  in  the  official  fore- 
casting formula  were  pointed  out,  and  the  method  of 

correlation,  with  the  equation   (y  —  y)  =  r  —  (x  -  x), 

was  shown  to  give  better  results  in  all  of  the  represent- 
ative states  and  in  qyqyy  month  of  the  growing  season. 
If  this  better  forecasting  formula  were  applied  to  the 
official  data  referring  to  the  condition  of  the  growing 
crop,  would  not  the  forecasts  of  yield  per  acre  be  better 
than  the  forecasts  which  we  have  been  able  to  make 
from  the  current  records  of  temperature  and  rainfall  in 
the  cotton  states? 

(2)  Although  data  as  to  the  condition  of  the  growing 
cotton  crop  have  been  officially  collected  and  pub- 
lished since  1866,  the  official  Bureaus  refrained,  until 
1911,  from  interpreting  their  own  data  in  the  definite 
form  of  quantitative  forecasts.  May  not  the  defects 
in  the  official  forecasts  which  we  have  located  and  meas- 
ured be  due  to  the  fact  that  the  official  prediction 
formula,  which  was  promulgated  in  1911,  has  been 
applied  in  this  Essay  to  data  running  through  a  cjuarter 
of  a   century? 

(3)  The  problem  of  measuring  the  relation  between 
the  yield  per  acre  of  cotton,  and  the  amount  of  rainfall 
and  the  temperature  at  various  epochs,  in  its  period 
of  growth,  presents  theoretical  and  practical  difficulties 


122     Forecasting  the  Yield  and  the  Price  of  Cotton 

that  leave  any  attempt  at  solution  open  to  possil)le 
objections.  The  chief  difficulties  are  (1)  that  the  series 
of  available  data  —  the  yield  per  acre  series  and  the 
series  of  rainfall  and  temperature  records  —  are  sum- 
mary results  of  three  classes  of  changes  with  three 
different  sets  of  causes:  (a)  secular  changes,  (b)  cyclical 
changes,  (c)  random  changes;  and  (2)  that  there  is  no 
known  statistical  method  that  will  enable  one  with 
series  as  short  as  ours  to  segregate  satisfactorily  the 
effects  of  these  three  types  of  changes.  May  it  not  be 
true,  therefore,  that  the  good  results  which  we  have 
obtained  in  our  forecasts  from  the  weather  are  results 
that  do  not  rest  upon  real  causes  but  are  largel^^  spu- 
rious, inhering  in  the  method  which  we  have  employed? 
We  shall  proceed  to  the  consideration  of  these  three 
objections.  Table  20  presents  the  data  that  are  neces- 
sary to  compare  the  accuracy  of  the  forecasts  from  the 
official  material  as  to  the  condition  of  the  crop,  and  the 
forecasts  from  the  records  of  temperature  and  rainfall, 
both  of  the  forecasts  being  made  by  the  methods  of 
correlation.  S  measures  the  accuracy  of  the  forecast 
from  the  condition  of  the  crop,  where  S  =  cr^\/l  —  r'-, 

and  the  forecasting  equation  is  (y  ~  y)  =  r  —  {x  —  x). 

0"., 

»S"  measures  the  accuracy  of  the  forecast  from  the 
weather  and  has  the  same  value  as  it  has  retained 
throughout  the  investigations  of  this  chapter.  The 
smaller  the  values  of  S,  S",  the  more  accurate  are  the 
respective  forecasts.  An  examination  of  Table  20 
shows  that  since  there  are  four  representative  states 
and  five  months  of  the  growing  season,  there  are  20 
cases  in  which  the  values  of  *S  and  S"  may  be  com- 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports  123 

pared.  In  12  out  of  these  20  cases  S"  is  smaller  than  S. 
For  purposes  of  exploiting  the  prospects  of  the  crop, 
the  earlier  a  reliable  forecast  can  be  obtained,  the 
greater  is  its  economic  value.  If,  therefore,  we  omit 
the  month  of  September,  there  are,  in  Table  20,  16 
cases  in  which  »S  and  *S"  may  be  compared,  and  in  12 
of  these  16  cases  *S"  is  sjnaller  than  S. 

TABLE  20.  — •  Relati\  E  Accuracy  of  the  Forecasts  of  the  Yield 
Per  Acre  of  Cotton  (1)  from  Data  as  to  the  Condition"  of 
the  Crop,  by  Means  of  the  Correlation  Equation;  (2)  prom 
Data  as  to  the  Weather,  by  Means  of  the  Method  of  Multi- 
ple Correlation 


Representative 

States 

Error  of  the  Forecast  from  the  Condition  of  the  Crop,  .S 
Error  of  the  Forecast  from  the  Weather,  S" 

May 

June 

.July 

August 

September 

5 

S" 

.S 

S" 

S 

S" 

S 

S" 

.S" 

S" 

Texas 

24.61 

2.5.31 

22.02 

25.15 

17.98 

22.58 

17.35 

16.80 

13.45 

16.80 

Georgia 

13.87 

12.67 

13.14 

11.28 

10.43 

9.94 

10.39 

9.46 

9.30 

9.46 
9.19 

Alabama 

13.36 

10.40 

12.02 

9.70 

10.93 

9.52 

10.44 

9.19 

9.19 

f-outh  Carolina 

18.64 

17.42 

17.38 

16   10 

15 .  .35 

16.10 

17.62 

14.08 

13 .  53 

14 .  98 

We  conclude  from  these  comparisons  that,  as  far  as 
concerns  the  representative  states  producing  sixty- 
five  per  cent  of  the  total  cotton  crop,  "the  forecasts 
from  the  weather  are  at  least  as  good  as  the  forecasts 
from  the  condition  of  the  crop  by  means  of  the  fonnula 

iy  —  y)  =  r  —  (.r  —  x).     This  latter  formula,  we  have 

already  seen,  gives  a  better  forecast  than  the  official 
formula  in  all  of  the  months  of  the  growing  season  of 
cotton,  in  all  of  the  representative  states. 


Table  21  contains  the  material  by  means  of  which  an 
opinion  may  be  formed  as  to  the  proper  answei-  to  the 


124     Forecasting  the  Yield  and  the  Price  of  Cotton 

second  of  our  three  questions.  Throughout  this  Essay 
the  error  of  the  ofhcial  formula  has  been  measured  by 

rwYiY^-^ic^)-} 

the  scatter  of  the  forecasts,  >S'=  y    -j Y  , 

and  in  case  of  each  of  the  representative  states,  the 
computed  value  of  S'  rested  upon  the  observations 
for  the  21  years,  1894  to  1914.  In  1911  the  Depart- 
ment of  Agriculture  defined  its  formula  for  pre- 
dicting, from  the  monthly  condition  of  the  crop,  the 
ultimate  yield  per  acre  of  cotton;  but  until  that  year 
the  condition  figures  were  published  without  an  official 
attempt  to  suggest  what  definite  inference,  as  to  the 
ultimate  yield,  should  be  made  from  the  crop  reports. 
It  might  therefore  be  assumed  that  the  year  1911  would 
mark  the  beginning  of  a  more  accurate  crop-reporting 
service,  and  that,  under  the  new  procedure  of  the  De- 
partment of  Agriculture,  the  conclusions  which  we 
have  drawn  concerning  the  comparative  accurac}-  of 
forecasts  from  the  condition  of  the  crop  by  means  of 
the  official  formula,  and  forecasts  from  the  weather 
by  means  of  correlation  equations,  would  no  longer 
hold  true.  Or  one  might  urge  that  throughout  the  en- 
tire period  1894-1914  the  crop-reporting  service  had 
been  continuously  improved,  and  that,  consequently, 
the  precision  of  the  official  formula  when  measured  by 
the  record  for  this  long  period  would  be  no  index  of  its 
present  accuracy.  These  very  natural  doubts  raise  two 
important  questions  of  fact:  (a)  Has  the  continuous 
improvement  of  the  crop-reporting  service  throughout 
the  21  years  1894-1914  been  such  that  the  error  of  the 
forecasts  since  1911  is  less  than  the  error  which  we  ob- 
tain when  the  official  formula  is  applied  to  the  data 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports  125 


5;   H 


o 

H     •- 
c-t    "^ 

>H      Pi 


<;    I 

a  en 


o  a 
^  a 
3  ^ 


~3 

C^l 

CO 

lO 

Ci 

v; 

" 

}- 

!J 

S: 

^" 

X 

CO 

o 

tC 

V- 

•f. 

^- 

l-O 

^ 

o 

tr^ 

v: 

rH 

'    ' 

„ 

*o 

-t< 

L'^ 

X 

V} 

" 

'Ji 

3 

^"' 

CC 

lO 

00 

C5 

3 

v^ 

1— 1 

»-H 

1~^ 

'"^ 

^_  1 

Cl 

^ 

, 

c: 

V; 

1-H 

1-H 

'""' 

'"^ 

_  ?.  rt 

C ;  CiO 

^" 

r- 

'f 

X 

Ci 

-^ 

"^ 

1  ^  ^J_ 

C0  35  05 

■/!       T 

X 

^;i 

Gj 

Ol 

o 

:o 

2   ^   ti 

rS  S   S 
S  O   a 

►^ 

ic 

"^ 

^ 

^^ 

*"* 

>■.  >■  >■ 

CD  ^  aj 

I 

^J=x: 

*j  -*^  -fc^ 

^" 

o 

!M 

TO 

X 

o^  o 

yj 

'— ' 

'— ' 

o  g  o 

1 

aaa 

y^ 

^^ 

^ 

X 

S 

C3 

lO 

t^ 

■o 

5 

y; 

T— ' 

'■"' 

■-5 

ro 

>o 

lO 

o 

v; 

iM 

" 

" 

■M 

» 

!M 

o 

X 

o 

v: 

iM 

'" 

>, 

TO 

o 

-+< 

,^ 

ti; 

v^ 

l>l 

(M 

T— 1 

C^l 

f. 

t^ 

t^ 

X 

:^i 

^■C 

C^l 

(M 

3'£ 

§3 

c3 

03 
§ 

5.1 

X 

^ 

o 

"^ 

O 

^ 

O 

120     Forecasting  the  Yield  and  the  Price  of  Cotton 

from  1894  to  1911?  (b)  Is  it  true  that  the  year  1911 
begins  a  period  of  marked  improvement  in  the  crop- 
reporting  service;  or,  in  more  definite  terms,  is  the 
error  of  the  forecasts  for  the  years  since  1911  less  than 
the  error  for  the  same  length  of  time  preceding  1911? 

Table  21  supplies  the  material  for  a  decision.  We 
recall  that  S'  is  the  coefficient  that  measures  the  error 
of  the  forecasts  for  the  whole  period  1894-1914.  S[,  S'^,, 
Ss,  in  Table  21,  have  the  following  meanings:  S[  is 
the  error  of  the  forecasts  when  the  official  formula 
is  applied  to  the  data  for  the  17  years  1894-1910;  S'^ 
is  the  error  of  the  forecasts  for  the  years  1911-1914; 
aSj  is  the  error  for  the  preceding  four  years,  1907-1910. 
We  shall  consider  first  the  comparative  values  of  S[,  S^. 

In  Texas,  the  forecasts  from  the  data  for  May, 
June,  August,  and  September  show  that  S[  is  in  all  four 
months  greater  than  S!,;  for  July,  the  two  coefficients 
are  equal.  Taking  the  values  of  these  coefficients  as 
they  are,  without  regard  to  their  probable  errors,  it  is 
legitimate  to  infer  that  in  Texas,  during  the  years 
1911-1914  as  compared  with  the  17  years  1894-1910, 
there  has  been  an  improvement  in  the  crop-reporting 
service. 

With  regard  to  Georgia  the  contrary  inference  must 
be  made.  In  three  out  of  five  cases  S[  is  less  than  S!,, 
while  in  the  remaining  two  cases  the  coefficients  are 
equal. 

In  Alabama,  there  has  possibly  been  some  improve- 
ment. In  three  out  of  five  cases  S[  is  greater  than  Si: 
in  one  case  ^i'  is  less  than  S!,;  and  in  one  case  the  two 
coefficients  are  equal. 

In  South  Carolina  there  has  been  no  change.     In 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports   127 

two  out  of  five  cases  *Si'  is  less  than  »S'._!;  in  two 
cases  S[  is  greater  than  S!,;  and  in  one  case  they  are 
equal. 

From  these  findings  the  conclusion  may  be  drawn 
that,  as  far  as  the  representative  states  are  concerned, 
there  has  been  no  such  improvement  in  the  crop- 
reporting  service  during  the  21  years,  1894-1914,  as  to 
make  questionable  the  testing  of  the  accuracy  of  the 
official  forecasting  formula  by  its  application  to  the 
data  which  we  have  actually  employed. 

We  come  now  to  the  other  question  of  fact:  Is  it 
true  that  the  year  1911,  when  the  Department  of  Agri- 
culture published  its  forecasting  formula,  initiated  a 
period  of  a  more  accurate  crop-reporting  ser\'ice? 
The  comparative  values  of  S',,  S'.^  give  the  necessary 
figures.  S!,  measures  the  accuracy  of  the  prediction 
for  the  four  years  1911-1914,  and  iSj  measures  the  ac- 
curacy of  the  forecasts  when  the  official  formula  is 
applied  to  the  data  of  the  four  years  preceding  1911, 
namely,  to  the  years  1907-1910.  There  are  20  cases  in 
which  the  values  of  S!,,  S'^  may  be  compared,  and  we 
find,  to  our  surprise,  that  in  15  out  of  20  possible  cases 
S'^  is  less  than  S!,;  that  is  to  say,  there  has  been  abso- 
lutely no  improvement  in  the  recent  crop-reporting 
service. 

It  cannot  reasonably  be  maintained,  therefore,  that 
because  of  improvements  in  the  official  forecasting 
service,  the  inferences  which  we  have  drawn,  concern- 
ing the  comparative  accuracy  of  the  forecasts  from  the 
weather  and  the  official  foi-ecasts  from  the  laboriously 
collected  data  about  crop  conditions,  have  not  an 
abiding  value. 


128     Forecasting  the  Yield  and  the  Price  of  Cotton 


Q 

y, 

< 

y. 

o 

o 

r-i 

a 

H 

■ji 

O 

Ui 

o 

a 

|4 

o 

^ 

H 

« 

z 

O 

o 

<^ 

H 

^ 

-< 

.«5 

O 

a 

c; 

M 

,^ 

a 

n 

a 

s 

x 

H 

o 

r> 

z 

K 

w 

H 

a 

a 

f^ 

z 

a 
rr; 

o 

a 

H 

a 

< 

a 

<A 

Q 

« 

z 

O 

a 

O 

> 

fe 

n 

^ 

J 

r/j 

a 

a 

<j 

a 

a 

2^ 

Z 

W 

<; 

P 

Ph 

r^ 

<; 

X 

a 

H 

. 

.^ 

X 

o 

-* 

c2 

i^ 

-* 

**2 

o 

1-H 

■M 

^ 

fi 

1 

ii 

a 

ri 

fO 

ro 

,_( 

X 

o  o 

fO 

(M 

CO 

a  s: 

'"! 

^ 

iM 

S  "*" 

H 

' 

' 

"^ 

^H 

O 

C^ 

^ 

CI 

X 

"3 

o 

s 

(- 

-iw 

a 

, 

<5 

ri 

X 

(M 

o 

c 

O    O 

M 

t^ 

.2 

^  '•=. 

O 

I^ 

1^ 

rt 

:j 

H 

1 

1 

1 

L^ 

0 

O 

— 

h- 

05 

'f 

-3 

lO 

o 

o 

o 

"i 

o 

»-* 

-M 

c 

*S 

l' 

o 

E 

^ 

o 

-3 

■* 

Ci 

t^ 

o 

t- 

I^ 

C^l 

o 

li 

^+1 

LQ 

lO 

~ 

:; 

"o 

r^ 

- 

~ 

"^ 

>o 

1— 1 

CO 

~zi 

35 

CO 

o 

'?3 

3 

►~5 

rj 

t^ 

O 

lO 

i- 

-+ 

Xi 

o 

t-2 

o 

o 

Oj 

t-, 

" 

'^ 

t- 

?o 

C^ 

o 

"O 

C^l 

C 

c 

1— 1 

■^ 

^ 

S' 

,-'. 

i* 

tT 

^ 

f— 1 

■■o 

■^  ?:^ 

to 

*o 

o 

•M 

—  — ' 

o 

H 

i 

2    9 

'^      . 

3 

cu    3 

^•2 

15 

O     0) 

fferenc 
acre, 
rainfa 

>^ 

o~  '3 

TS     4J     oT 

^ 

^      1^ 

a  S 

J  -2" 

22  "^ 

"o  2 

"x    ii    js 

>-i 

^ 

a^ 

C   3c 


o    3 


O     t,    ^ 

s,(i  o  *^  ■;;    c 
"-i    ,    X  ~^  i: 


P  ^' 


bC    o5 

51 


o    cj  .=:  ■:3 


—    V    fi    t^ 


_    ti    «^    ?    t: 
"^3  IB    o    3 

i  ^   i:    -2    "J; 

000;=!— 


i:   c    c    iJ   2 


"  8  i 


5  "< 
?  '^ 


O 


t;  '^  "i 

"i  'i'  "S 

X 

^    .,  -3 


r- 


>i  c<i  CO 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports   129 

Table  22  presents  important  calculations  bearing 
upon  the  adequacy  of  the  method  which  we  have  em- 
ployed to  measure  the  relation  between  the  yield  per 
acre  of  cotton  and  the  variations  in  temperature  and 
in  rainfall.  As  has  been  already  suggested,  the  statis- 
tics of  the  yield  per  acre  of  cotton  and  the  records  of 
temperature  and  rainfall  are  the  summary  expressions 
covering  results  of  three  classes  of  changes  — ■  secular, 
cyclical,  and  random  changes  —  with  three  different 
sets  of  causes.  Moreover  the  series  which  we  ha\'e 
made  the  basis  of  our  investigation  cover  only  a  quarter 
of  a  century.  Our  problem  is  to  discover  the  true 
relations  between  the  weather  and  the  yield,  to  the 
end  that  the  knowledge  of  the  natural  relations  may  be 
made  a  basis  of  a  reliable  system  of  forecasting  the  yield. 
But  we  are  at  once  confronted  with  a  dilemma.  If  we 
employ  the  best  method  for  measuring  the  true  rela- 
tion between  two  variables  each  of  which  is  expressed 
in  a  numerical  series  made  up  of  secular,  cyclical,  and 
random  changes,  we  need  very  long  series  of  observa- 
tions, whereas  our  own  particular  series  are  relatively 
short;  if,  on  the  other  hand,  we  apply  a  smipler  method 
to  our  limited  but  complex  series,  our  findings  are  liable 
to  be  spurious  in  the  sense  of  resulting  from  the  in- 
adequacy of  the  method  which  we  have  employed  and 
not  resting  upon  natural  relations  of  the  variables.  The 
best  we  can  do  under  the  circiunstances  is  to  compare 
the  results  of  applying  different  methods  to  the  same 
problem,  and  if  we  find  that  with  the  most  api:)roved 
devices  there  is  an  agreement  in  the  essential  results, 
we  are  justified  in  an  accession  of  faith  in  our  work. 

The  query  that  naturally  presents  itself  with  refer- 


130     Forecasting  the  Yield  and  the  Price  of  Cotton 

eiice  to  the  method  which  we  have  adopted,  in  this 
chapter,  of  correlating  ratios  —  the  ratios  being  de- 
rived from  progressive  averages  of  three  years  — ■  is 
whether  the  correlations  that  we  obtain  are  true  cor- 
relations in  the  sense  of  indicating  the  presence  of  real 
causes,  and  not  spurious  correlations  having  their  origin 
in  some  singularity  of  the  method  itself.  Until  very 
recently  statistical  science  offered  no  means  of  treating 
adequately  the  difficulty  that  confronts  us,  and,  indeed, 
where  the  series  to  be  compared  are  relatively  short, 
there  is  even  now  no  entirely  satisfactory  method 
for  ascertaining  the  true  relation  of  the  variables.  But 
the  Variate  Difference  Correlation  Method  which  has 
been  gradually  elaborated  by  Professor  Pearson  ^  and  his 
co-workers,  is,  as  far  as  I  am  aware,  the  method  that  is 
freest  from  theoretical  objections  when  it  is  applied 
to  such  limited  series  as  we  are  compelled  to  work 
with.  If,  therefore,  a  comparison  of  the  results  of  the 
method  of  correlating  ratios  with  the  results  of  the  ap- 
plication of  the  Variate  Difference  Method  shows  a 
substantial  agreement  in  the  signs  and  magnitudes 
of  the  coefficients,  then  the  force  of  one  of  the  chief 
objections  to  our  work  is  greatly  reduced. 

In  order  to  make  a  test  case,  we  shall  take  the  cor- 
relation of  the  yield  per  acre  of  cotton  and  the  records 
of  temperature  and  rainfall  in  Texas.  The  reason  for 
selecting  Texas  is  because  we  have  found  that  in  Texas 
alone,  among  the  representative  states,  the  forecasts 
from  the  weather  are  not  in  all  cases  better  than  the 
forecasts  from  the  condition  of  the  crop.    In  Texas,  only 

1  Biomeirika,  Vol.  X,  Parts  II  and  III,  Xovember.  1914,  pp.  340-3.')o, 
and  the  references  there  cited. 


Forecasting  the  Yield  of  Cotton  from  Weather  Reports   131 

two  of  the  five  months  give  forecasts  from  the  weather 
that  are  better  than  the  forecasts  from  the  condition 
of  the  crop,  while  in  all  of  the  other  states,  for  every 
month,  the  forecast  from  the  weather  is  better  than  the 
forecast  from  the  condition  of  the  crop.  A  comparison, 
therefore,  of  the  results  in  Texas,  in  the  manner  which 
we  have  proposed,  will  have  the  advantage  not  only  of  a 
test  of  our  method  but  will  also  settle  the  question 
whether,  by  the  use  of  a  different  method,  we  might 
not  obtain  for  Texas  the  uniformly  better  forecasts 
from  the  weather  which  we  have  obtained  for  the  other 
representati\'e  states. 

An  examination  of  the  computations  in  Table  22 
shows  that,  when  attention  is  given  to  the  probable 
errors  of  the  coefficients,  there  is  a  remarkable  agree- 
ment between  the  correlations  of  ratios  and  the  cor- 
relations of  first  differences  and  of  second  differences. 
Two  essential  points  are  brought  out  by  the  three  rows 
of  coefficients:  (1)  All  three  series  indicate  that  the 
critical  factors  in  the  growth  season  of  cotton,  in  Texas, 
are  the  July  temperature,  the  August  temperature,  and 
the  August  rainfall;  (2)  The  method  that  we  have 
adopted  in  this  chapter  to  measure  the  relation  be- 
tween cotton  yield  and  the  elements  of  the  weather 
does  not,  in  this  test  case,  exaggerate  the  degree  of  asso- 
ciation between  the  variables. 

From  the  results  of  this  test  case  we  draw  the  im- 
portant conclusion  that  the  method  of  correlating 
ratios  does  enable  us  to  discover  the  critical  factors  in 
the  growth  of  the  cotton  plant;  and  that  the  forecasts 
based  upon  the  correlations  of  ratios  ai'e  not  spuiious. 
but  rest  u]:)on  real  causes. 


APPENDIX 


TABLE  23.  —  Texas.  Official  Monthly  Reports  on  the  Con- 
dition OF  THE  Growing  Cotton  Crop,  and  Official  Final 
Estimates  of  the  Annual  Yield  Per  Acre  in  Pounds  of 
Cotton  Lint 


Year 

Condition  of  the  Crop 

Yield  per 

Acre  in 

Pounds  of 

Lint 

June  1 

July  1 

August  1 

September  1 

October  1 

1889 
1890 

1 

9.5 

90 

91 

81 

78 

169 

84 

89 

82 

77 

77 

190 

91 

95 

92 

82 

78 

195 

2 

81 

87 

86 

81 

77 

291 

3 

82 

84 

72 

•   63 

65 

151 

4 

94 

99 

85 

84 

88 

235 

5 

79 

70 

71 

56 

58 

151 

6 

92 

SO 

69 

62 

.57 

104 

7 

87 

88 

78 

70 

64 

165 

8 

89 

92 

91 

75. 

73 

212 

9 

90 

93 

87 

61 

56 

185 

1900 

71 

78 

83 

77 

78 

226 

1 

84 

86 

74 

56 

51 

1.59 

2 

9.5 

73 

77 

53 

47 

14S 

3 

70 

79 

82 

76 

54 

143 

4 

84 

89 

91 

77 

69 

183 

.5 
C 

69 

72 

71 

70 

69 

164 

87 

82 

86 

78 

74 

■225 

7 

8 

9 

1910 

11 

12 

13 

14 

70 

72 

75 

67 

60 

130 

77 

80 

82 

75 

71 

196 

78 

79 

70 

59 

52 

125 

83 

84 

82 

69 

63 

145 

88 

85 

86 

68 

71 

186 

86 

89 

84 

76 

7.5 

206 
150 
184 

84 

S6 

81 

64 

63 

65 

74 

71 

79 

70 

Appendix 


133 


TABLE  24.  —  Georgia.  Official  Monthly  Reports  on  the  Con- 
dition OF  THE  Growing  Cotton  Crop,  and  Official  Final 
Estimates  of  the  Annual  Yield  Per  Acre  in  Pounds  of 
Cotton  Lint 


Year 

Condition  of  the  Crop 

Yield  per 

Acre  in 

Pounds  of 

Lint 

June  1 

July  1    August  1 

September  1 

October  1 

18S9 
1S90 

SO 

80 

91 

90 

87 

155 

94 

95 

94 

80 

82 

165 

1 
2 

SO 

85 

86 

82 

78 

1.55 

S7 

SS 

84 

79 

75 

160 

3 

87 

SO 

83 

77 

7(") 

136 

4 
•5 

76 

78 

85 

84 

79 

155 

82 

88 

87 

70 

72 

152 

6 

9.5 

94 

92 

71 

07 

122 
178 

7 

S4 

85 

95 

SO 

70 

8 

89 

90 

91 

so 

75 

183 

9 

88 

85 

79 

09 

04 

07 

159 

1900 

89 

74 

77 

09 

172 

1 

SO 

72 

78 

SI 

73 

107 

2 

94 

91 

83 

OS 

02 

165 
15S 

3 
4 
•5 

75 

75 

77 

81 

OS 

78 

85 

91 

SO 

78 

205 

84 

82 

82 

77 

70 

200 

ti 
7 
8 
9 
1910 
U 
12 
13 

1-^ 

SO 

82 

74 

72 

OS 

165 

74 

78 

SI 

SI 

76 

190 

80 

83 

85 

77 

71 

190 

84 

79 

78 

73 

71 

184 

81 

78 

70 

71 

68 

173 

92 

94 

95 

81 

79 

240 

74 

72 

08 

70 

65 

163 

69 

74 

76 

76 

72 

20S 

SO 

S3 

S2    1     SI 

81 

239 

184     Forecasting  the  Yield  and  the  Price  of  Cotton 


TABLE  25.  —  Alabama.    Official  Mo.nthly  Repouts  on  the  (.'o\- 

DITION     OF    THE     GROWING     CoTTON     CrOP,     AND     OFFICIAL    FiNAL 

Estimates  of  the   Annual   Yield   Per   Acre   in   Pounds   of 
Cotton  Lint 


Year 

Condition  of  the  Crop 

Yield  per 

Acre  in 

Pounds  of 

Lint 

.June  1 

July  1 

August  1 

September  1 

October  1 

1889 

83 

87 

90 

91 

87 

163 

1890 
1 
2 
3 
4 
•5 

93 

95 

93 

84 

80 

160 

89 

87 

89 

S3 

76 

165 

91 

90 

83 

72 

()9 

135 

82 
88 

80 

79 

78 

76 

148 

87 

94 

86 

84 

160 

8.5 

83 

81 

71 

70 

135 

6 

103 

98 

93 

66 

61 

124 

7 

81 

85 

88 

80 

73 

1.55 

8 
9 
1900 
1 
2 

89 

91 

95 

80 

76 

195 

86 

88 

82 

76 

70 

176 

87 

70 

67 

64 

62 

151 

76 

80 

82 

75 

65 

1.56 

92 

84 

77 

54 

52 

144 
161 

3 
4 

73 

76 

79 

84 

68 

80 

85 

90 

84 

76 

182 

.5 

87 

83 

79 

70 

70 

173 

6 

81 

84 

83 

76 

68 

165 

7 

65 

68 

72 

73 

68 

169 

S 

78 

82 

85 

77 

70   ■ 

179 
142 

9 

83 

64 

68 

66 

62 

1910 
11 

83 

81 

71 

72 

67 

160 

91 

93 

94 

80 

73 

204 

12 
13 

74 

76 

73 

75 

68 

173 

75 

79 

79 

72 

67 

190 

14 

85 

88 

81 

77 

78 

209 

Appendix 


135 


TABLE  26.  —  South  Carolina.  Official  Monthly  Reports  on 
THE  Condition  of  the  Growing  Cotton  Crop,  and  Official 
Final  Estimates  of  the  Annual  Yield  Per  Acre  in  Pounds 
OF  Cotton  Lint 


Year 

Condition  of  the  Crop 

Yield  per 

Acre  in 

Pounds  of 

Lint 

June  1 

July  1 

August  1 

September  1 

October  1 

1889 

78 

84 

90 

87 

81 

141 

1890 

97 

95 

95 

87 

83 

175 

1 

SO 

80 

83 

81 

72 

160 

2 

91 

94 

83 

77 

73 

184 

3 

88 

83 

75 

63 

62 

142 

4 
■5 

83 

88 

95 

SO 

79 

168 

72 

84 

81 

82 

64 

141 

() 

97 

98 

88 

70 

67 

129 

7 

87 

86 

92 

84 

74 

189 

8 
9 

8.5 

90 

89 

81 

79 

245 

80 

88 

78 

66 

62 

165 

1900 
1 

8.5 

79 

74 

60 

.57 

167 

80 

70 

75 

SO 

67 

141 
199 

2 

97 

9.5 

88 

74 

68 

3 

76 

74 

70 

SO 

70 

178 

4 

81 

88 

91 

87 

81 

215 

5 

78 

78 

79 

75 

74 

220 

6 

82 

77 

72 

71 

66 

175 

7 

77 

79 

81 

S3 

77 

215 

8 

9 

1910 

81 

84 

84 

76 

68 

219 

S3 

77 

77 

74 

70 

210 

78 

7.5 

70 

73 

70 

216 
280 

11 

80 

84 

80 

74 

73 

12 

83 

79 

7.5 

73 

68 

209 

13 

68 

73 

75 

77 

71 

235 

14 

72 

81 

79 

77 

72 

2.55 

13G     Forecasting  the  Yield  and  the  Price  of  Cotton 


TABLE  27.  —  Texas.     Tempekatuhi.:   (Dec; 
Rainfall  (Inches)  in  Eastern  and 


RKE.S    FaHKE.NHEIT)     AXD 

Central  Texas 


Year 

May 

June 

July 

August 

•September 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

1891 

71.3 

2.64 

81.7 

2.48 

82.7 

1.71 

81.1 

1.70 

77.3 

2.26 

2 

73.5 

4.55 

79.5 

4.37 

82.8 

1.85 

80.9 

4.12 

75.5 

1.40 

3 

73.0 

4.71 

80.0 

2.88 

85.0 

0.75 

81.6 

2.19 

79.3 

1.79 

4 
.5 

74.5 

3.00 

78.8 

2.56 

82.0 

2.03 

79.8 

5.26 

76.9 

2.57 

71.3 

7.01 

78.6 

6.18 

82.8 

4.11 

83.5 

1.78 

80.5 

1.79 

6 

S 
9 

77.8 

1.62 

83.4 

0.99 

84.8 

1.94 

85.2 

1.64 

78.0 

4.56 

72.3 

4.33 

80.6 

3.65 

85.9 

1.26 

83.0 

2.32 

76.9 

2.58 

74.5 

3.55 

80.2 

5.63 

82.3 

2.09 

82.6 

2.50 

77.7 

1.61 

77.1 

3.49 

79.8 

6.56 

82.5 

2.00 

86.6 

0.36 

76.7 

1.16 

1900 

72.3 

5.89 

81.8 

1.85 

81.8 

4.50 

82.0 

2.98 

80.8 

6.60 

1 

72.4 

4.22 

81.8 

1.08 

85.4 

1.99 

85.2 

1.77 

76.8 

3.27 

2 

76.4 

3.78 

82.8 

2.19 

81.6 

7.73 

85.2 

0.11 

75.0 

4.51 

3 

70.0 

2.27 

73.9 

3.70 

81.3 

5.91 

82.4 

1.71 

74.7 

2.98 

4 
5 

72.3 

4.76 

79.4 

4.57 

81.8 

2.45 

81.9 

2.16 

79.0 

2.82 

74.9 

5.95 

81.1 

4.47 

80.7 

4.90 

83.9 

1.03 

79.5 

2.02 

6 

72.6 

3.96 

80.5 

3.83 

80.8 

5.13 

80.8 

4.46 

78.3 

3.50 

7 

67.6 

6.80 

80.4 

1.85 

83.1 

2.87 

85.1 

1.01 

79.0 

1.29 

8 

73.2 

7.87 

81.3 

2.19 

81.8 

2.60 

82.2 

2.28 

76.0 

3. .54 

9 

72.2 

2.91 

81.1 

3.07 

86.5 

1.56 

85.4 

2.00 

78.0 

0.88 

1910 
11 

71.4 

4.04 

80.2 

1.79 

84.8 

1.15 

86.4 

0.83 

81.5 

1.78 

73.2 

1.50 

84.7 

0.65 

82.8 

4.36 

84.6 

2.93 

83.1 

1.53 

12 

74.0 

2.21 

77.4 

3.28 

85.4 

1.04 

84.0 

3.47 

77.9 

0.77 

13 
14 

73.0 

3.55 

79.0 

2.66 

84.9 

1.52 

84.7 

1.03 

73.2 

5.70 

71.1 

7.81 

82.4 

1.31 

86.2 

0.91 

80.7 

8.95 

77.4 

1.39 

Appendix 


13- 


TABLE  28.  —  Geokgia.    Temperature  (Degrees  Fahrenheit)  and 
Rainfall  (Inches) 


Vear 

M 

a\- 

June 

Ju 

b- 

Aus 

ust 

.September     1 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

1892 

72.1 

2.16 

78.1 

5.93 

78.1 

6.13 

76.3 

6.18 

72.4 

3.61 

3 

70. 5 

2.04 

74.9 

4.53 

81.2 

2.78 

78.3 

6.63 

75.0 

2.86 

4 
.5 

6 

71.4 

2.51 

77.1 

2.72 

78.1 

7.82 

78.5 

5.20 

75.1 

3.72 

69.8 

4.20 

77.9 

3.93 

79.0 

4.96 

79.1 

7 .55 

77.0 

1.53 

70.0 

2.54 

77.7 

3.55 

80.0 

8.26 

81.6 

2,89 

76,3 

2.37 

7 

69.3 

1.52 

80.8 

3.51 

80.6 

5.74 

78.3 

5.07 

73.8 

2.83 

8 

74.0 

1.12 

80.1 

3.27 

79.8 

8.14 

78.5 

10.09 

75.3 

4,76 

9 

1900 

75.2 

1.76 

80.2 

2.59 

80.2 

4.53 

81.1 

4.58 

73.4 

1.40 

70.8 

2.46 

75 . 6 

8.98 

80.3 

5.12 

82.3 

2 .55 

77.3 

3.06 

1 

71.4 

5.71 

77.6 

5.26 

81.5 

4.18 

78.2 

9.92 

73.1 

5,19 

2 
3 

75.5 

2.34 

79.5 

3.54 

82.0 

4. 55 

80.3 

3.92 

73.0 

4.67 

70.3 

5.47 

74.4 

6.00 

80.0 

4.00 

80.7 

5.56 

73.1 

4.40 

4 

70.2 

2.23 

77.4 

2.95 

79.0 

3.81 

77.6 

7.33 

76.0 

1.48 

5 

0 

74.5 

5.02 

78.7 

3.69 

80.3 

5.57 

78.6 

4,96 

76.8 

2,95 

70.2 

4.32 

77.8 

6.31 

77.8 

8.41 

80.1 

5.82 

77.5 

5,29 

7 

70.1 

4.26 

76.0 

4.29 

Sl.O 

5.04 

79.5 

4.10 

75. 5 

6,24 

S 

72.1 

2.67 

77.5 

3.40 

79.8 

5.27 

79.0 

6.04 

73.2 

3.11 

9 
1910 

69.7 

4.43 

78.5 

5.48 

79.0 

5.25 

79.8 

4.53 

74.2 

3.17 

69.8 

3.61 

75  3 

7.16 

78.9 

5.68 

79.1 

3.68 

76.3 

2.48 

11 
12 

73.1 

2.14 

80.9 

2.78 

78.3 

5.44 

79.4 

6.18 

79.6 

2  95 

72.6 

4.08 

75.4 

6.83 

79.5 

5.71 

79.1 

4  91 

77.3 

5.73 
4.12 

13 

71.9 

2.27 

76.4 

4.83 

81.3 

5.61 

79.3 

4.03 

72.3 

14 

72.7 

0  74 

82,2 

3  51 

SO.  8 

4  74 

79.1 

5.99 

72  7 

3  53 

138     Forecasting  the  Yield  and  the  Price  of  Cotton 


TABLE  29.  —  Alabama.      Temperature     (Degrees    Fahrenheit) 
AND  Rainfall  (Inches) 


Year 

May 

June 

Jul.v 

Augu.st 

September 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

1890 

75.8 

3.44 

77.2 

5.24 

80.9 

5.09 

82.2 

2.30 

75.8 

1.76 

7 

68.6 

1.56 

80.9 

1.85 

81.1 

4.78 

78.8 

5.58 

75.6 

0.55 

8 

73.6 

0.82 

80.4 

3.60 

80.0 

6,06 

78.7 

7.43 

75.5 

3.58 

9 

75.9 

2.03 

79.8 

2,54 

80.4 

6.76 

81.3 

3.6^ 

72.7 

0.66 

1900 

71.2 

2.64 

76.4 

11.08 

79.8 

4.93 

81.6 

2.89 

77.8 

4,00 

1 
2 

69.8 

5.08 

78.5 

2.80 

82.2 

3.40 

78.6 

8.86 

72.1 

4.19 

75.4 

2.34 

80.8 

1.28 

82.8 

2.50 

82.1 

3.48 

73.4 

4.28 

3 

69.6 

6.05 

73.2 

4.88 

80.0 

3.98 

80.5 

3.57 

73.2 

1.42 

4 

69.6 

2.98 

77.8 

2.94 

79.6 

4,80 

78.4 

5.55 

76.8 

1.36 

5 

74.2 

5.51 

79.0 

4.56 

79.4 

4.56 

79.2 

5.30 

76.2 

2.51 

(5 

69.7 

4.63 

78.9 

3,45 

78.8 

8.50 

80.4 

3.78 

78.2 

8.44 

7 

68.0 

7,94 

75.6 

2,85 

81.0 

5.00 

80.4 

3.50 

74.8 

5.50 

8 

71.4 

5.34 

77.5 

2.75 

79.8 

4.72 

79.4 

3.44 

74.2 

2.42 

9 

68.6 

6.51 

78.0 

7.82 

79.3 

4.52 

81.0 

3.30 

73.7 

2.87 

1910 

68.9 

3.86 

75.6 

6,98 

78.6 

7.18 

79.7 

2.73 

77.5 

2.21 

11 

72.9 

2.85 

80.6 

3.86 

78.0 

5.66 

79.1 

4.97 

80.4 

2.32 

12 
13 

72.0 

3.60 

75.1 

5.10 

79.7 

5.17 

79.2 

5.68 

77.1 

4.79 

71.6 

3.14 

77.5 

3.54 

81.1 

5.00 

80.5 

2.58 

73.1 

6.96 

14 

71.8 

1.05 

83.1 

2.66 

81.6 

4.23 

79.1 

6.41 

72.4 

4.69 

1.5 

74.5 

6.34 

78.8 

3.66 

80.4 

5.23 

78.9 

5.07 

76.6 

4.43 

Appendix 


139 


TABLE  30.  —  South  Carolina.    Temperature  (Degrees 
Fahrenheit)  and  Rainfall  (Inches) 


Year 

Ma.\- 

June 

July 

August 

September 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

Tem- 
pera- 
ture 

Rain 
fall 

Tem- 
pera- 
ture 

Rain- 
fall 

1S91 
2 

69.0 

3.57 

79.3 

3.20 

77.9 

5.95 

78.9 

8.79 

74.3 

2.66 

71.1 

5.53 

75.1 

5.25 

78.9 

7.44 

79.5 

4.42 

72.5 

6.41 

3 

70.2 

4.13 

76.5 

7.64 

82.0 

3.87 

77.5 

12.45 

74.7 

4.42 

■4 
5 

70.7 

3.43 

77.0 

3.91 

77.7 

8.24 

77.9 

7.28 

75.0 

6.51 

69.0 

4.36 

78.2 

3.04 

79.5 

4.17 

79.4 

7.95 

76.9 

1.29 

0 

7 

76.7 

2.74 

77.9 

5.42 

80.7 

8.17 

80.4 

4.14 

75.0 

2.94 

69.3 

2.39 

79.2 

5.44 

80.2 

5.01 

78.0 

5.16 

73.3 

2.91 

S 
9 

73.8 

1.35 

79.7 

4.15 

80.0 

7.81 

78.7 

9.81 

76.0 

4.06 

73.7 

1.68 

79.4 

3.89 

80.0 

4.03 

81.2 

6.26 

72.8 

2.55 

1900 

1 
2 

70.2 

2.37 

76.2 

7.94 

81.2 

4.08 

83.0 

2.13 

77.1 

2.83 

71.4 

7.31 

76.7 

6 .  55 

81.4 

4.52 

78.6 

9.01 

73.1 

4.66 

74.0 

2.69 

78.5 

4.48 

80.8 

3.79 

78.6 

5.07 

72.1 

3.74 

.3 

70.7 

2.69 

74.2 

8.09 

80.4 

3.59 

80.6 

7.15 

72.7 

3.62 

4 

70.6 

2.04 

77.0 

4.06 

79.4 

5.96 

77.6 

8.47 

75.8 

2.46 

.5 

73.4 

5.70 

78.9 

1.92 

80.4 

6.16 

77.9 

5.69 

76.2 

1.91 

6 

70.7 

3.00 

78.4 

8.88 

78.4 

8.40 

80.6 

6.62 

78.0 

4.85 

7 

70.8 

4.51 

75.8 

5.92 

81.4 

5.06 

79.4 

5.41 

76.0 

5.91 

S 

71.8 

2.92 

76.6 

4.90 

79.8 

5.43 

78.6 

9.11 

72.4 

2.86 

9 
1910 

69.5 

4.26 

79.2 

6.87 

78.6 

4.92 

78.8 

4.83 

72.0 

3.74 

69.8 

4.03 

75.5 

7.78 

79.4 

5.83 

79.0 

6.00 

75.8 

3.10 

11 
12 
13 
14 

72.6 

0.65 

80.9 

3.42 

79.8 

3.79 

80.0 

6.05 

78.9 

3. 33 

72.4 

4.08 

75.5 

5.68 

79.6 

5.22 

79.2 

3.69 

77.7 

5.91 

71.7 

2.13 

76.2 

5.53 

81.9 

4.78 

78.9 

3.76 

71.7 

4.66 

72.1 

0  83 

81   1 

3.80 

80.0 

5.56 

79.0 

5 .  88 

71.3 

3.63 

CHAPTER  V 

THE  LAW  OF  DEMAND  FOR  COTTON 

"There  is  a  general  agreement  as  to  the  character  and  directions  of 
the  changes  which  various  economic  forces  te^id  to  produce.  .  .  . 
Much  less  progress  has  been  made  towards  the  quantitative  determina- 
tion of  the  relative  strength  of  different  economic  forces." 

—  Alfred  Marshall. 

The  investigations  of  the  preceding  two  chapters 
have  made  us  acquainted  with  the  degree  of  reUabiUty 
of  the  Government  reports  on  the  prospective  cotton 
crop  and  with  the  measure  of  accuracy  with  which,  at 
any  stage  in  the  growth  season,  the  prospective  yield 
of  cotton  may  be  calculated  from  the  past  conditions 
and  vicissitudes  of  the  weather.  The  new  problem  that 
we  face  in  this  chapter  carries  the  inquiry  to  its  final 
stage:  Assuming  that  the  ultimate  volume  of  the  crop 
may  be  forecast  with  a  known  degree  of  precision,  is  jt 
possible  to  predict  the  relation  that  will  subsist  between 
the  size  of  the  crop  and  the  price  of  cotton  lint?  Is  it 
possible  to  know  the  dynamic  law  of  the  demand  for 
cotton? 

Two  Practical  Methods  of  Approach 

In  Chapters  III  and  IV,  we  found  that  the  method  of 
progressive  averages  enabled  us  to  get  valuable  results 
in  the  problem  of  forecasting  the  amount  of  production 
from  the  Government  reports  on  the  condition  of  the 
growing  crop,  and  from  the  records  of  temperature  and 
rainfall  in  the  Cotton  Belt.     We  shall  test  the  helpful- 


The  Law  of  Demand  for  Cotton  141 

ness  of  this  same  device  in  our  present  inquiry  as  to  the 
form  of  the  concrete  law  of  demand  for  cotton. 

Method  of  progressive  averages.  In  Table  31  the  data^ 
are  collected  for  computing  the  relation  between  the 
price-ratio  and  the  production-ratio  of  cotton.  The 
problem  to  be  solved  may  be  put  into  symbolic  form: 
Let  p  be  the  mean  price  per  pound  of  cotton  for  any 
given  year,  and  p^  be  the  mean  price  for  the  preceding 
three  years;  let  P  be  the  total  production  of  cotton  for 
the  given  year,  and  F3  be  the  mean  production  for  the 
preceding  three  years.  Our  problem  is  to  find  (1)  the 
coefficient  of  correlation  measuring  the  relation  between 
Pfpz  and  P IPi]  (2)  the  statistical  law  connecting  P'tp^ 
with  P/Ps,  which  is  the  concrete  law  of  demand  for 
cotton;  (3)  the  error  incurred  in  using  the  law  of  de- 
mand for  cotton  as  a  formula  with  which  to  forecast 
the  price  of  cotton  from  the  prospective  size  of  the  crop. 

The  values  of  the  series  P/ps  and  P/Ps,  for  the  period 
1890  to  1913,  are  given  in  columns  4  and  7  of  Table  31. 
The  calculation  of  the  items  that  constitute  the  solu- 
tion of  our  problem  gives: 

(1)  The  coefficient  of  correlation  between  P/ps  and 
P/Ps  is  r  =  -  .706; 

>  The  crude  data  are  taken  from  the  Statistical  Abstract  of  the  United 
States,  1914,  p.  505.  "The  pro(kiction  statistics  n>late,  when  possible, 
to  the  year  of  growth,  but  when  figures  for  the  j'ear  are  wanting,  a  com- 
mercial crop  which  re}>resents  the  trade  movement  is  taken.  The  sta- 
tistics of  production  have  been  compiled  from  publications  of  the 
United  States  Department  of  Agriculture  for  181)0  to  1S9S.  Census 
figures  have,  however,  been  used  when  available,  including  those  for 
1899  to  date."    Ibid.,  note  1. 

"The  value  of  lint  per  pound  shown  since  1902  relates  to  the  average 
grade  of  upland  cotton  marketed  \n-ior  to  April  1  of  the  following  year; 
from  1890  to  1901,  the  average  price  of  middling  cotton  on  the  New 
Orleans  Cotton  Exchange."    Ibid.,  note  2. 


142     Forecasting  the  Yield  and  the  Price  of  Cotton 


TABLE  31. 


The    Phoduction-Ratio    and    the    Piuck-Hatkj    of 
Cotton 


Year 

Efiuivalent 
501)  Pound 
Bales,   Gros.- 

Weight 

(Millions  of 

Bales) 

P 

Mean  Pro- 
duction for 
the  Preced- 
ing Three 
Years 

Pz 

Production- 
Ratio 

n?. 

Price  per 

Pound 

Upland 

Cotton 

(Cents) 

P 

Mean  Price 

for  the 

Preceding 

Three 

Years 

Pi 

Price- 
Ratio 

1  Pi 

1887 

6.88 

10.3 

8 

6.92 

10.7 

9 

7.47 

11.5 

1890 

1 

8.56 

7.09 

120.7 

8.6 

10.8 

79.6 

8.94 

7.65 

116.9 

7.3 

10.3 

70.9 

2 

6.66 

8.32 

80.0 

8.4 

9.1 

92.3 

3 

7.43 

8.05 

92.3 

7 .5 

8.1 

92.6 

4 

10.03 

7.68 

130.6 

5.9 

7.7 

76.6 

5 

7.15 

8.04 

88.9 

8.2 

7.3 

112.3 

6 

8.52 

8.20 

103.9 

7.3 

7.2 

101.4 

7 

10.99 

8.57 

128.2 

5.6 

7.1 

78.9 

8 

11.44 

8.89 

128.7 

4.9 

7.0 

70.0 

9 

9.35 

10.32 

90.6 

7.6 

5.9 

128.8 

1900 

10.12 

10.59 

95.6 

9.3 

6.0 

155.0 

1 

9.51 

10.30 

92.3 

8.1 

7.3 

111.0 

2 

10.63 

9.66 

110.0 

8.2 

8.3 

98.8 

3 

9.85 

10.09 

97.6 

12.2 

8.5 

143.5 

4 

13.44 

10.00 

134.4 

8.7 

9.5 

91.6 

5 

10.58 

11.31 

93.5 

10.9 

9.7 

112.4 

6 

13.27 

11.29 

117.5 

10.0 

10.6 

94.3 

7 

11.11 

12.43 

89.4 

11.5 

9.9 

116.2 

8 

13.24 

11.65 

113.6 

9.2 

10.8 

85.2 

9 
1910 

10.00 

12.54 

79.7 

14.3 

10.2 

140.2 

11.61 

11.45 

101.4 

14.7 

11.7 

125.6 

11 
12 

15.69 

11.62 

135.0 

9.7 

12.7 

76.4 

13.70 

12.43 

110.2 

12.0 

12.9 

93.0 

13 

14.16 

13.67 

103.6 

13.1 

12.1 

108.3 

The  Law  of  Demand  for  Cotton  143 

(2)  The  concrete  law  of  demand  for  cotton  h  y  = 

—  .975.T  +  206.03;  where  x  is  put  for  P'lPz,  and  y  is  the 
most  probable  value  of  ^Ipz,  corresponding  to  the  given 
value  of  Pi  Pi) 

(3)  The  accuracy  with  which  the  law  of  demand  for 
cotton  may  be  used  to  forecast  the  price  of  cotton  lint 
is  measured  by  >S  =  o-^v/l  —  r''  =  16.38. 

Method  of  percetdage  changes.^  In  Table  32  the  crude 
statistical  data  of  production  and  prices  are  utilized 
in  a  different  way.  From  year  to  year  both  the  price 
and  the  production  of  cotton  undergo  changes,  and  in 
the  construction  of  Table  32  the  hypothesis  in  mind 
suggested  that  there  is  a  close  relation  between  the  per- 
centage change  of  the  price  in  any  given  year  over  the 
price  of  the  preceding  year,  and  the  percentage  change 
in  production  of  the  given  year  over  the  production  of 
the  preceding  year.  The  percentage  changes  are  tabu- 
lated in  columns  4  and  7. 

The  calculations  based  upon  the  data  of  this  Table 
show  that 

(1)  The  coefficient  of  correlation  between  the  per- 
centage change  in  price  and  the  percentage  change  in 
production  is  r  =  —  .819; 

(2)  The  dynamic  law  of  demand  for  cotton  is  y  = 

-  1 .08.T  +  8.81 ;  where  x  is  put  for  the  percentage  change 
in  production,  and  y  is  the  most  probable  value  of  the 
percentage  change  in  price,  corresponding  to  the  gi\'en 
percentage  change  in  production; 

(3)  The  accuracy  with  which  the  dj^namic  law  of 

'  A  more  ample  descriijtion  of  this  nietliod  is  contained  in  Ecoxoniic 
Cycles:  Their  Law  and  Cause,  Chai)ter  I\'. 


144     Forecasting  the  Yield  and  the  Price  of  Cotton 

demand  for  cotton  may  be  used  to  forecast  the  percent- 
age change  in  the  price  of  cotton  lint  is  measured  by 

s  =  o-,vr^^'  =  15.18. 


TABLE  32.  —  Percentage  Changes  in  the  Price  and  Production 
OF  Cotton  Lint 


Year 

Equivalent 

500  Pound 

Bales,  Gross 

Weight 

(Millions  of 

Bales) 

Change 

over  the 

Preceding 

Year 

Percentage 

Change 

over  the 

Preceding 

Year 

Price  per 
Pound 
Upland 
Cotton 

(Cents) 

Change 

over  the 

Preceding 

Year 

Percentage 

Change 

over  the 

Preceding 

Year 

1889 

7,47 

11.5 

1890 

8.56 

+  1.09 

+  14.59 

8.6 

—  2.9 

—  25 . 22 

1 
2 

S.94 

+  0.38 

+    4.44 

7  3 

—  1.3 

—  15   12 

6.66 

—  2.28 

—  25 . 50 

8.4 

+  1.1 

+  15.07 

3 

7.43 

+  0.77 

+  11.56 

7.5 

—  0.9 

—  10.71 

4 

10.03 

+  2.60 

+  34 . 99 

5.9 

—  1.6 

—  21.33 

5 

7.15 

—  2.88 

—  28.71 

8.2 

+  2.3 

+  38.98 

0 

8.52 

+  1 .  37 

+  19.16 

7.3 

—  0.9 

—  10 . 98 

7 

10.99 

+  2.47 

+  28 . 99 

5.6 

—  1.7 

—  23  29 

8 

11.44 

+  0.45 

+    4.10 

4.9 

—  0.7 

—  12.50 

9 

9.35 

—  2.09 

—  18.27 

7.6 

+  2.7 

+  55 . 10 

1900 

10.12 

+  0.77 

+    8.24 

9.3 

+  1.7 

+  22.37 

1 

9.51 

—  0.61 

—    6.03 

8.1 

—  1.2 

—  12.90 

2 

10.63 

+  1.12 

+  11.78 

8.2 

+  0.1 

+    1.23 

Z 

9.85 

—  0.78 

—    7 .  34 

12.2 

+  4.0 

+-4S.7S 

4 

13.44 

+  3.59 

+  36.45 

8.7 

—  3.5 

—  28.69 

5 

10.58 

—  2.86 

—  21.28 

10.9 

+  2.2 

+  25.29 

6 

13.27 

+  2 .  69 

+  25.43 

10.0 

—  0  9 

—    8.26 

7 

11.11 

—  2.16 

—  16.28 

11.5 

+  1.5 

+  15.00 

8 

13 .  24 

+  2.13 

+  19.17 

9.2 

—  2.3 

—  20.00 

9 
1910 

10.00 

—  3.24 

—  24.47 

14.3 

+  5.1 

+  55.43 

11.61 

+  1.61 

+  16.10 

14.7 

+  0.4 

+    2.80 

11 

15.69 

+  4.08 

+  35.14 

9.7 

—  5.0 

—  34.01 

12 

13.70 

—  1.99 

—  12.68 

12.0 

+  2.3 

+  23.71 

13 

14.16 

+  0.46 

+    3  36 

13.1 

+  1.1 

+    9.17 

The  Law  of  Demand  for  Cotton  145 

Figure  11  makes  clear  to  the  eye  the  measure  of  agree- 
ment between  the  actual  percentage  changes  in  price 
and  the  percentage  changes  as  they  are  predicted  from 
the  law  of  demand. 

A  comparison  of  the  results  we  obtain  from  these 
two  methods  of  deriving  the  law  of  demand  for  cotton 
shows  that  there  is  very  little  difference  between  them 
so  far  as  the  accuracy  of  the  forecasts  are  concerned. 

But,  in  Chapter  I.  we  have  said  that  it  is  possible 
to  forecast  the  price  of  cotton  from  the  size  of  the  crop 
with   greater  accuracy   than   the  Bureau   of  Statistics 
can  forecast  the  yield  of  cotton  from  the  known  condi- 
tion of  the  growing  crop.    This  statement  we  shall  now 
prove.     Throughout  our  investigations  we  have  meas- 
ured   the    accuracy    of    forecasts    by    >S  =  o'^n/i  —  r"-, 
where  a,^  is  the  standard  deviation  of  a  concrete  series, 
and  r  is  the  correlation  between  two  series.    *S  measures 
the  accuracy  of  the  forecasts  because  it  shows  how  the 
prediction  formula  enables  one  to  reduce  their  vari- 
ability.     If   there    were   no   forecasting    formula    the 
variability  of  the  series  that  we  wish  to  know  would 
be  (Ty,  but  by  the  use  of  the  formula  the  variability 
of  the  forecasts  is  only  cr,,  \/l  -  /•'-•    The  factor  \/l  —  r- 
measures  the  reduction  in  variability  that  is  gained  by 
means  of  the  forecasting  formula.    If,  therefore,  we  wish 
to  compare  the  accuracy  of  forecasts  of  two  different 
series,  the  measure  of  the  relative  accuracy  is  given  by 
\/l  —  r',   and  the   smaller  the  value  of  \/l  —  r',   the 
greater  the  accuracy  of  the  forecasts.     The  same  idea 
may  be  put  in  a  different  way  by  saying  that  the  greater 
the  value  of  r,  the  greater  the  accuracy  of  the  forecasts. 


146     Forecasting  the  Yield  and  the  Price  of  Cotton 


uo^o:)yo  33ud  31/j.  u/  s'3£uc>L/^  sSd^u^:?^^^ 


fo 


The  Law  of  Demand  for  Cotton  147 

If  now  we  refer  to  Chapter  III,  "The  Government 
Crop  Reports,"  we  find  that  the  correlation  between 
the  predicted  yield-ratio  of  cotton  and  the  actual  yield- 
ratio  is,  for  the  month  of  May,  r  =  —  .049;  for  June, 
r  =  .292;  for  July,  r  =  .595;  for  August,  r  =  .576;  for 
September,  r  =  .685.  But  the  calculations  of  the  pres- 
ent chapter  have  shown  that  the  correlation  between 
the  price-ratio  of  cotton  lint  and  the  j^roduction-ratio 
is  r  =  —  .706;  and  the  correlation  between  the  per- 
centage change  of  prices  and  the  percentage  change  of 
production  is  r  =  —  .819. 

Statics  and  Dynamics  Discriminated 

The  law  of  demand  for  cotton,  in  the  form  in  which 
it  was  treated  in  the  preceding  section,  was  deduced 
from  official  descriptions  of  changes  extending  over  a 
quarter  of  a  century.  We  have  found  that  the  correla- 
tion between  the  percentage  change  of  prices  and  the 
percentage  change  of  production  is  r  =  —  .819;  if  we 
had  used  the  data  extending  as  far  back  as  1869,  when 
the  disastrous  effects  of .  the  Civil  War  upon  prices 
had  not  yet  been  overcome,  we  should  have  found  r  = 
—  .736,  which  is  still  a  high  coefficient.  Our  law  of 
demand  is  a  dynamic  law;  it  is  a  summary  description 
of  a  routine  in  concrete  affairs. 

The  law  of  demand  in  statical  economics  is  of  a 
different  quality,  and  the  hypothetical  limitations  of 
deductions  based  upon  it  are  not  always  kept  in  mind. 

According  to  Professor  Marshall,  "there  is  .  .  .  one 
general  law  of  demand:  The  greater  the  amount  to  be 
sold,  the  smaller  must  be  the  price  at  which  it  is  offered 


148     Forecasting  the  Yield  and  the  Price  of  Cotton 

in  order  that  it  niaj^  find  purchasers;  or,  in  other  words, 
the  amount  demanded  increases  with  a  fall  in  price, 
and  diminishes  \yith  a  rise  in  price."  "The  one  univer- 
sal rule  to  which  the  demand  curve  conforms  is  that  it  is 
inclined  negatively  throughout  the  whole  of  its  length."  ^ 
This  statement  of  the  law  is  absolute.  Remembering 
the  explicit  claim  made  in  the  Preface  to  the  last  edi- 
tion of  Professor  Marshall's  work  (the  fifth  edition  of 
1907)  that  his  volume  is  "concerned  throughout  with 
forces  that  cause  movement :  and  its  key-note  is  that  of 
dynamics,"  one  might  fall  into  the  error  of  supposing 
that  the  deductions  based  upon  the  law  applied  directly 
to  actual  phenomena.  But  Professor  Marshall  has 
carefully  pointed  out  the  reservations  that  are  made: 

(1)  It  is  assumed  in  giving  definite  form  to  the  law  of 
demand  for  any  one  commodity  that  the  prices  of  all 
other  commodities  remain  constant.  This  is  the  usual 
cceteris  paribus  assumption.'-  With  regard  to  this 
hypothesis  I  should  like  to  quote  the  comment  of  Pro- 
fessor Marshall's  sympathetic  fellow-worker.  Professor 
Edgeworth:  "Demand  curves  as  usually  understood 
involve  a  postulate  which  is  frequently  not  fulfilled; 
namely,  that  while  the  price  of  the  article  under  con- 
sideration is  varied,  the  prices  of  all  other  articles  re- 
main constant.  This  postulate  fails  in  the  case  of 
rival  commodities  such  as  beef  and  mutton.  The  price 
of  one  of  these  cannot  be  supposed  to  rise  or  fall  con- 
siderably without  the  price  of  the  other  being  affected. 
The  same  is  true  of  commodities  for  which  there  is  a 
joint-demand  as  for  malt  and  hops.    And  in  case  of  a 

>  Marshall:  Principles  of  Ecotiotuics,  5th  ed.,  p.  99,  noto  2. 
-  Ibidem,  pp.  x,  100. 


The  Law  of  Demand  for  Cotton  149 

necessary  of  life  the  price  cannot  be  supposed  to  increase 
indefinitely  without  the  prices  of  other  articles  falling, 
owing  to  the  retrenchment  of  expenditure  on  articles 
other  than  necessaries."  "It  is  true,  indeed,  that  the 
postulate  which  has  been  stated  might  be  dispensed 
with.  But  this  can  only  be  done  at  the  sacrifice  of 
two  of  the  characteristic  advantages  which  demand 
curves  offer  the  theorist.  First,  unless  this  postulate 
is  granted,  it  is  hardly  conceivable  that,  when  the  prices 
of  several  articles  are  disturbed  concurrently,  the  col- 
lective demand  curve  may  be  predicted  by  ascertain- 
ing the  disposition  of  the  individual  —  a  conception 
which  aids  us  to  apprehend  the  working  of  a  market. 
Secondly,  when  the  prices  of  all  commodities  but  one 
are  not  supposed  fixed,  there  no  longer  exists  that 
exact  correlation  between  the  demand  cur\'e  and  the 
interests  of  consumers  in  low  prices  which  Prof.  Mar- 
shall has  formulated  as  'consumers'  rent.'"  ' 

(2)  The  validity  of  the  law  is  limited  to  a  point  in 
time.-  Refering  to  this  limitation,  Professor  Edge- 
worth  remarks:  ''There  is  an  artificial  rigidity  in  de- 
mand curves  which  imperfectly  correspond  to  the  flux 
character  of  human  desires.  One  cause  of  change  is  the 
formation  of  new  habits.  The  increased  use  of  petro- 
leum is  not  to  be  ascribed  simply  to  the  fall  in  price, 
the  demand  curve  being  supposed  constant,  but  rather 
to  the  fact  that  'petroleum  and  petroleum  lamps  have 
become  familiar  to  all  classes  of  society'  (Marshall)."  '^ 

'  Palgrave's  Dictionurij  of  Polltiail  Kconoiin/,  \'o\.  I.  ''Demand 
Curves,"  pp.  543-544. 

Principles  of  Economics,  pp.  U4,  100. 

•^Palgrave's  Dictiommj  of  Political  Econoimj.  \u\.  I,  ''Deniaiid 
Curves,"  p.  544. 


150     Forecasting  the  Yield  and  the  Price  of  Cotton 

(3)  The  usual  statement  of  the  law  of  demand  does 
not  take  "account  of  the  fact  that,  the  more  a  person 
spends  on  anything  the  less  power  he  retains  of  pur- 
chasing more  of  it  or  of  other  things,  and  the  greater 
is  the  value  of  money  to  him  (in  technical  language 
every  fresh  expenditure  increases  the  marginal  value 
of  money  to  him)."  ^ 

(4)  When,  however,  account  is  taken  of  the  varying 
marginal  utility  of  money,  it  is  possible  that  the  de- 
mand curve  for  food,  on  the  part  of  the  "poorer 
labouring  famihes,"  shall  be  positively  incUned. 
But  in  the  statement  of  the  law  of  demand,  Pro- 
fessor Marshall  has  said  "the  one  universal  rule  to 
w^hich  the  demand  curve  conforms  is  that  it  is  inclined 
negatively."  - 

(5)  "Again,  the  demand  for  a  commodity  on  the 
part  of  dealers  who  buy  it  only  with  the  purpose  of 
selling  it  again,  though  governed  by  the  demand  of  the 

'  Principles  of  Economics,  p.  132. 

-  Ibidem,  pp.  132,  99  note. 

Referring  to  Professor  Pareto's  recognition  of  the  theoretical  possi- 
bihty  of  obtaining  curves  of  demand  of  the  positive  type,  M.  Zawadzki 
makes  the  extremely  valuable  criticism  which  I  have  italicized  in  the 
following  quotation:  "Quelle  est  la  valeur  d'une  telle  conclusion? 
N'est-eUe  pas  en  contradiction  flagrante  avec  les  faits?  II  est  facile 
d'imaginer  des  cas  theoriques  ou  la  demande  diminuerait  a  la  suite 
d'une  diminution  de  prix.  La  theorie  doit  done  etre  capable  d'en  tenir 
compte.  Ont-ils  lieu  en  realile  (et  dans  Vhypothese  statique)  autrement 
que  par  exception?  Quelle  pent  etre  leur  importance?  Void  des  questions, 
et  on  pourrait  en  poser  hien  d'autres,  nuxquelles  la  theorie  ne  nous  repond 
pas.  Dans  cet  exemple  nous  touchons,  pour  ainsi  dire,  du  doigt  la  puis- 
sance et  la  faiblesse  de  V economic  mathematique .  Nous  avons  la  formule 
la  plus  generale,  englobant  jusqu'd  des  cas  extremement  rares,  jnais  nous 
ne  pouvons  pas  en  passer  aux  cas  particuliers,  pas  mime  distinguer  ce 
qui  est  I'exception  de  ce  qui  est  la  regie."  Les  Mathematiques  Appliqnees 
a  U Economic  Politique,  p.  186. 


The  Law  of  Demand  for  Cotton  151 

ultimate    consumers    in    the    background,    has    some 
pecuharities  of  its  own."  ^ 

(6)  The  hope  of  obtaining  concrete,  statistical  laws 
of  demand  was  expressed  by  Jevons  in  1871,  and  has 
been  repeated  by  Professor  Marshall  in  the  successive 
editions  of  his  Principles  from  1890  to  1907.  But  ac- 
cording to  Professor  Edgeworth  ''.  .  .  it  may  be 
doubted  whether  Jevons's  hope  of  constructing  de- 
mand curves  by  statistics  is  capable  of  realisation."  - 

A  Co77iplete  Solution  of  the  Problem 

The  listing  of  the  reservations  that  are  made  by 
Professor  ]Marshall  when  he  states  "the  one  universal 
rule  to  which  the  demand  curve  conforms"  has  the 
double  advantage  of  cautioning  his  reader  against 
drawing  pi-ecipitate  conclusions  as  to  the  applicability 
to  concrete  affairs  of  any  theoretical  deductions  based 
upon  the  curve,  and  of  suggesting  the  imperati\'e  need 
of  a  more  concrete  treatment  of  the  law  of  demand. 
With  the  derivation  of  demand  curves  from  statistical 
data  we  shall  be  concerned  in  the  present  section. 

We  shall  be  aided  in  approaching  our  problem  if  we 
put  it  into  symbolic  form:  Suppose  we  let  .To  be  the 
percentage  change  in  the  price  of  a  conmiodity,  say, 
for  instance,  cotton,  and  let  Xi  be  the  percentage  change 
in  the  amount  of  the  conmiodity  that  is  demanded. 
Then,  if  my  interpretation  of  Professor  Marshall's 
view  is  correct,  his  understanding  of  the  nature  of  the 
law  of  the  demand  may  be  described  in  two  stages: 

1  Marshall:  Principles  of  Economics,  \).  100  n. 

-  Palgrave's  Dictionary  of  Political  Econotnij,  \\)\.  I,  "Demand 
Curves,"  p.  .544. 


152    Forecasting  the  Yield  and  the  Price  of  Cotton 

(1)  In  reality  .To  =  (/)Cti,  X2,  Xo, . .  .xj,  where  X2,  Xs, 
. .  .x,^  are  percentage  changes  in  other  factors,  some 
of  which  are  enumerated  in  Professor  Marshall's  ex- 
plicit reservations  when  he  formulates  the  law  of  de- 
mand. The  form  of  the  function  <^  is  unknown  and  the 
interrelations  of  Xi,  X2,  Xs, . . .  x„  are  unknown. 

(2)  In  the  statement  of  the  law  of  demand  in  its 
absolute  form,  Xi  is  singled  out  as  the  important  variable 
in  relation  to  Xo,  and  the  law  of  demand  in  its  static 
form  expresses  the  relation  that  exists  between  Xo  and 
Xi  when  .To,  Xs, . . .  x,^  are  all  equal  to  zero.  These 
variables  X2,  Xs, . .  .x„  must  be  equal  to  zero  since  they 
severally  represent  percentage  changes,  and  the  general 
hypothesis  in  mind  when  the  static  law  of  demand  is 
formulated  is  that  there  shall  be  no  changes  in  other 
economic  factors. 

The  misgivings  that  one  feels  about  conclusions  which 
are  based  upon  the  static  law  of  demand  are  due  to  the 
fact  that  the  form  of  the  function  (f)  is  not  known ;  that 
the  influence  of  the  factors  X2,  X3,...x„  is  ignored; 
and  that  the  interrelations  of  To,  Xs, . . .  x„  have  not 
been  determined.  The  misgivings  would  be  removed 
if  these  three  limitations  could  be  overcome. 

The  procedure  that  I  wish  to  introduce  —  the  treat- 
ment of  the  problem  statistically  by  the  method  of 
multiple  correlation  —  addresses  itself  precisely  to  these 
three  limitations.  The  ultimate  aim  of  economic  theory 
is  to  enable  us  to  forecast  economic  phenomena,  and, 
in  this  particular  problem  of  the  law  of  demand,  we 
wish  to  forecast  Xo,  the  percentage  change  in  the  price. 
We  know  that  To  =  <^(^i,  ^2,  ^3,  • .  -x,,),  and  while  we 
do  not  know  either  the  form  of  the  function  <^,  or  the 


The  Law  of  Demand  for  Cotton  153 

interrelations  of  Xi,  X2,  Xs,  .  ■ .  x„,  the  practical  prob- 
lem l^efore  us  suggests  effectual  means  of  overcoming 
these  Imiitations.  For,  since  our  ultimate  object  is 
to  forecast  the  value  of  Xo  and  to  measure  the  degree 
of  accuracy  with  which  the  forecast  is  made,  we  may, 
with  due  precautions  against  spurious  results,  experi- 
ment with  different  types  of  the  function  <fi  and  of  the 
interrelations  of  Xi,  X2,  X3,  .  .  .i'„,  and  settle  upon  those 
types  that  enable  us  to  forecast  Xo  with  a  degree  of 
precision  sufficient  for  the  actual  problem  in  hand. 

As  a  first  approximation  we  naturally  take  the 
simplest  type  of  functions.     We  say,  suppose 

(1)  That  the  tj^pe  of  the  function  <j)  is  linear,  such 
that 

(f)ixi.  x-2,  X:u  .  .  .  .r„ )  =  u  =  fl„+  fli.ri+  flo.r.>+  .  .  .+  r/„.T„ ; 

(2)  That  the  interrelations  of  ^'1,  .To,  x^,  .  .  .  x„  are 
also  linear,  such  that,  for  example  .Ti  =  61  +  b-ix-i. 
This  second  supposition  will  present  no  difficulties, 
since  we  have  dealt  with  the  problem  of  finding  the 
relation  between  .Ti  and  X2  when  the  connection  is  of 
the  simple  form  .ri  =  61  +  ^o-r-j.  With  regard  to  the 
first  supposition,  all  that  we  need  to  do  in  order  to  ob- 
tain a  satisfactory  practical  solution  of  our  problem  is 
so  to  determine  from  the  actual  statistics  the  values 
of  the  constants  a,,,  ai,  ...a„  that  the  correlation  be- 
tween .To  and  u,  which  we  designate  by  R,  shall  be  a 
maximum.  In  that  case  the  value  of  S  =  cto \/l  —  R', 
which  measures  the  root-mean-square  error  of  the  fore- 
casts by  means  of  the  formula  Xu  =  flo  +  ai.Ti  +  a-yx-y 
H-  .  .  .  +  (i„x„,  will  be  a  minimum. 

Now  precisely  this  problem  has  already  recei\'ed  a 


154     Forecasting  the  Yield  and  the  Price  of  Cotton 

general  solution  in  the  statistical  theory  of  niuhij)le 
correlation.  If,  for  the  sake  of  simplicity,  we  take  the 
case  of  three  variables,  then  the  equation  connecting 
Xo,  X\j  X2.  in  such  a  way  that  the  correlation  is  a  maxi- 
mum between  the  actual  values  of  Xq  and  the  predicted 
values  of  Xq  has  the  form 

,         _  .     rill  —  ro2/'r2  ^11  -  \  1^''"'-     ^'ui''i2  o'o  /         _  s 

(Xn  —  Xo)  =      ;         7,  \X\      X\)~\        -        7,  [X'z      X2) 

l  —  r].^     0-1  1-^2     o"2 

r'in  -\-  r^o     Zrinr^^orvi 


and  S  =  aQ\/l  —  R-,  where  R-  = 


1       2 
1  —  ^12 


The  forecasting  equation  enables  us  to  predict  the  most 
probable  values  of  Xn  from  the  known  values  of  Xi,  X2, 
and  S  measures  the  degree  of  accuracy  with  which  the 
forecasts  are  made. 

An  example  will  make  this  abstract  discussion  much 
clearer.  Professor  Edgeworth  makes  the  statement 
that, 

"One  important  cause  of  alteration  in  demand 
curves  is  the  increase  of  the  consumer's  purchasing 
power.  The  case  in  which  that  increase  is  only  apparent, 
being  due  to  a  rise  in  prices  (and  the  converse  case), 
may  be  specially  distinguished.  Owing  to  the  variabil- 
ity, it  may  be  doubted  whether  Jevons's  hope  of  con- 
structing demand  curves  by  statistics  is  capable  of 
realisation."  ^ 

Professor  Edgeworth  doubtless  meant  that  because 
of  the  many  factors  tending  to  produce  a  variation  in 
the  demand  schedule  it  might  be  doubtful  whether 
Jevons's  hope  could  be  reahsed.     But  suppose  —  for 

1  Palgrave's  Dictionary  of  Political  Econotinj,  "Demand  Curves," 
Vol.  ],  p.  544. 


The  Law  of  Demand  for  Cotton  155 

the  sake  of  simplicity  and  concreteness  but  in  illustra- 
tion of  a  method  of  complete  solution  — ■  we  limit  our 
inquiry  to  this  question :  How  may  the  relation  between 
the  price  of  cotton  and  the  amount  of  cotton  demanded 
be  determined  (1)  when  account  is  taken  of  the  varying 
purchasing  power  of  money ;  (2)  when  there  is  no  varia- 
tion in  the  purchasing  power  of  money? 

Let  Xo  be  the  percentage  change  in  the  price  of 
cotton,  .ri  be  the  percentage  change  in  the  amount  of 
cotton  produced,  and  .r^  be  the  percentage  change  in  the 
index  number  of  general  prices.  We  may  then  put  our 
problem  and  its  solution  in  this  form: 

(1)  .To  =  (f)(.Vi,  Xo),  and  we  assume  as  a  preliminary 
hypothesis  that  the  form  of  <j)(xi,  x-^)  is  linear  so  that 
we  may  WTite  .ro  =  Wo  +  ciiXi  +  a-^Xn.  According  to  the 
theor}^  of  multiple  correlation,  when  the  values  of 
Oo,  ai,  a-2  are  so  determined  from  the  actual  data  as  to 
make  the  correlation  between  the  actual  values  of  Xo 
and  the  \'alues  of  .r,,  when  forecast  by  the  above  formula 
a  maximum,  then 

(.To  —  To)  =  —; -, (Ti  -  Xi)  H 7, (To  -  .To). 

1— r]o       (Ti  l—r'ri       (To 

The  statistical  material  necessary  for  the  computation 
of  the  quantities  indicated  in  these  symbols  is  given  in 
Table  33.  When  the  actual  computations  are  made 
and  the  numerical  values  are  substituted  in  the  abo\'e 
equation,  we  obtain  as  our  forecasting  formula 

.To=  -  .97ti+  l.()().r,+  7.11. 

This  formula  enables  us  to  j)redict  the  probable  value 
of  .To  for  given  values  of  Ti.  x->:  it  enables  us  to  say  what 


156     Forecasting  the  Yield  and  the  Price  of  Cotton 

TABLE  33.  —  Data  for  the  Quantitative  Detehminatiox  of  the 
Law  of  Demand  for  Cotton 


Year 

Equivalent 

.500  Pound 

Bales,  Gross 

Weight 

(Millions  of 

Bales) 

Price  per 

Pound  of 

Upland 

Cotton 

(Cents) 

Bureau  of 
Labor's  In- 
dex of  Prices 
of  "All  Com- 
modities" 

Percentage 

Change  in 

the  Amovint 

Produced 

Percentage 
Change  in 
the  Price 
of  Cotton 

Percentage 
in  the  In- 
dex of  Gen- 
eral Prices 
.r-j 

1889 

7.47 

11.5 

115 

1890 

8.56 

8.6 

113 

-1-14.59 

—  25.22 

—  1.74 

1 

S.94 

7.3 

112 

+    4.44 

—  15.12 

-0.88 

2 

f) .  66 

8.4 

106 

—  25 . 50 

+  15.07 

—  5.36 

3 

7.43 

7.. 5 

106 

+  11.56 

—  10.71 

0.00 

4 

10.03 

5.9 

96 

H-  34  99 

—  21.33 

—  9 .  43 

5 

7.15 

8.2 

94 

—  28.71. 

-)- 38.98 

—  2.08 

6 

8.. 52 

7.3 

90 

-1-19.16 

—  10.98 

—  4.26 

7 

10.99 

5.6 

90 

-h  28.99 

—  23.29 

0.00 

S 

11.44 

4.9 

93 

+    4.10 

—  12.50 

-r3.33 

9 

9.35 

7.6 

102 

—  18.27 

-1-55.10 

-i-9.68 

1900 

10.12 

9.3 

110 

+    8.24 

-1-22.37 

-!-  7 .  84 

1 

9.51 

8.1 

108 

—    6.03 

—  12.90 

—  1 .  82 

2 

10.63 

8.2 

113 

+  11.78 

+    1.23 

-1-  4 .  63 

3 

9.85 

12.2 

114 

—    7.34 

-h  48.78 

-}-  0 .  88 

4 

13.44 

8.7 

113 

-1-  36 .  45 

—  28.69 

-0.88 

5 

10.58 

10.9 

116 

—  21  28 

4-25.29 

-f  2.65 

6 

IS.  27 

10  0 

122 

+  25.43 

—    8.26 

-i-5.17 

7 

11.11 

11.5 

130 

—  16.28 

+  15.00 

-f  6.56 

S 

13.24 

9  2 

123 

+  19.17 

—  20.00 

—  5.38 

9 

10.00 

14.3 

126 

—  24 . 47 

-1-  55.43 

-1-2.44 

1910 

11.61 

14.7 

132 

-1-16.10 

-f-    2.80 

-1-4.76 

U 

15.69 

9.7 

129 

-f  35.14 

—  34.01 

—  2  27 

12 
13 

13.70 

12.0 

134 

—  12  68 

-h23  71 

-h  3 .  88 

14.16 

13.1 

135 

-H    3.36 

+    9.17 

-f-  0 .  75 

the  probable  change  in  the  price  of  cotton  will  be  when 
we  know  the  probable  changes  in  the  production  of 
cotton  and  in  the  level  of  general  prices.  Figure  12 
traces  for  a  period  of  twenty-fiA^e  years  the  actual 
variations  in  the  percentage  changes  in  the  price  of 


The  Law  of  Demand  for  Cotton 


157 


UQ^OJJO 


158     Forecastiitg  the  Yield  and  the  Price  of  Cotton 

cotton,  together  with  the  percentage  changes  as  the}'' 
are  predicted  by  means  of  the  formula  a-o  =  —  .97a- 1  + 
1.60x2  +  7.11. 

(2)  The  degree  of  accuracy  with  which  the  above 
forecasting  formula  enables  us  to  predict  the  changes 
in  the  price  of  cotton  is  measured  b}^ 

*S  =  o-(is/  1  —  R-,  where  R-  =  ; ^ . 

1  —  ^2 

The  computations  from  the  statistical  data  of  Table 
33  shows  that 

R  =  .859,  and  S  =  13.56. 

This  is  a  very  high  coefficient  of  correlation,  and  con- 
sequently the  forecasting  formula  makes  possible  the 
prediction  of  the  changes  in  the  price  of  cotton  with  a 
relatively  high  degree  of  precision. 

We  have  now  a  solution  of  the  first  part  of  our  prob- 
lem. We  know  how  to  forecast  the  changes  in  the  price 
of  cotton  when  account  is  taken  of  the  changes  in  the 
amount  demanded  and  of  variations  in  the  purchasing 
power  of  money.  We  know%  besides,  the  degree  of 
reliability  with  which  our  forecasts  are  made.  We  next 
enter  upon  the  second  part  of  our  problem:  What  is  the 
relation  between  the  changes  in  the  price  of  cotton  and 
the  changes  in  the  ajiiount  demanded  when  there  are 
no  changes  in  the  purchasing  power  of  money? 

(3)  Since,  in  the  forecasting  formula  Xo  =  —  .97xi 
+  1.60x2  +  7.11,  the  variables  Xo,  Xi,  X2  are  percentage 
changes,  if  we  put  X2  =  0,  we  obtain  the  answer  to  the 
second  part  of  our  problem.  The  equation  Xo  =  —  .97xi 
+  7.11  expresses  the  relation  between  the  changes  in 


The  Law  of  Demand  for  Cotton  159 

the  price  of  cotton  and  the  changes  in  the  amount  of 
cotton  demanded  ivhen  the  purchasing  power  of  money 
rernains  constant.  Figure  13  traces  the  course  of  the 
actual  changes  in  the  price  of  cotton,  and  the  changes 
as-  they  would  occur  under  the  supposition  that  the 
le\'el  of  general  prices  remains  constant.  The  root- 
mean-square  error  of  the  forecasts  by  means  of  this 
formula  is  >S  ==  15.38. 

Furthermore,  by  the  theory  of  iDartial  correlation 
we  know  that  when  .To  is  constant  —  in  this  case  when 
.To  =  zero  —  the  coefficient  measuring  the  relation  be- 

,        .  ''oi       Tdorjo 

tween  .To  and  .Ti  is  poi  =     /—         — /,         ,  ,  whicli,  bv 

V  1  —  ?7,o  V  1  —  r]., 

computation  from  the  statistical  data  of  Table  33,  gives 

Poi  =  -  .808. 

(4)  If  we  collect  our  results  bearing  upon  the  rela- 
tion of  changes  in  the  price  of  cotton,  changes  in  the 
amount  of  cotton  demanded,  and  changes  in  the  pur- 
chasing power  of  money,  we  find  that  we  have  con- 
sidered their  interrelations  under  three  different  aspects : 

(i)  The  relation  between  .To  and  .ri,  when  no  atten- 
tion is  paid  to  the  variable  To,  and  .I'n  is  regarded  as  a 
simple  function  of  Xi.  This  is  the  case  of  the  dynamic 
law  of  demand  in  its  simplest  form.  Here  roi  =  —  .819; 
S  ^  15.18.    The  graph  is  given  in  Figure  11. 

(ii)  The  relation  between  To  and  both  Ti  and  To,  where 
To  is  regarded  as  a  function  of  two  \'arial)les.  This  is 
illustrative  of  the  case  of  the  dynamic  law  of  demand  in 
its  complex  form. 

Here  R  =  .859;  .S  =  13.50.  The  graph  is  gi\-en  in 
Figure  12. 

(iii)  The  relation  between  t,,  and  Ti,  when  t^  =  0; 


160      Forecasting  the  Yield  and  the  Price  of  Cotton 


Zya//OJ  JO   3JUC/  3l/4  U/   S^SuOl^O   ji>O^U33J3cJ 


The  Laiv  of  Demand  for  Cotton  IGl 

that  is  to  say  the  relation  between  the  changes  in  the 
price  of  cotton  and  the  changes  in  the  amount  of  cot- 
ton demanded,  when  the  purchasing  power  of  money 
remains  constant.  This  is  illustrative  of  the  static 
law  of  demand. 

Here  pm  =  —  .808;  S  =  15.38.  The  graph  is  given 
in  Figure  13. 

A  consideration  of  these  results  will  show  how  theo- 
retical difficulties  disappear  before  a  practical  solution. 
One  of  the  discouraging  aspects  of  deductive,  mathe- 
matical economics  is  that  when  a  complete  theoretical 
formulation  is  given  of  the  possible  relations  of  factors 
in  a  particular  problem,  one  despairs  of  ever  arriving 
at  a  concrete  solution  because  of  the  multiplicity  of  the 
interrelated  variables.  But  the  attempt  to  give  statis- 
tical form  to  the  equations  expressing  the  interrelations 
of  the  variables  shows  that  many  of  the  hypothetical 
relations  have  no  significance  which  needs  to  be  re- 
garded in  the  practical  situation.  When  we  write  the 
law  of  demand  in  the  form  .To  =  (f>ixi,  X2,  Xs,  .  .  ..t„), 
it  is  true,  as  we  have  pointed  out,  that  we  do  not  know 
the  form  of  <f)  nor  the  types  of  the  interrelations  of  .ri,  x-z, 
X3,...x„;  but  when  we  are  confronted  with  the  prac- 
tical problem  of  forecasting  Xo,  we  can  find  empirical 
functions  that  enable  us  to  predict  .Xo  with  a  high  degree 
of  accuracy.  Nor  is  this  the  only  result  of  the  prac- 
tical solution.  We  find  that  since  Voi  =  ~  .819  and 
Pol  =  —  .808,  there  is  really  no  difference  in  the  close- 
ness of  the  relation  between  .ro  and  .ri  whether  we  com- 
pletely ignore  the  variations  in  the  purchasing  power 
of  money  or  regai'd  the  purchasing  power  of  money  as 


1()2     Forecasting  the  Yield  and  the  Price  of  Cotton 

constant.  And  since  R  =  .859,  we  learn  that  while 
the  relation  between  .To  and  .To  may  be  fairly  high  fin 
this  case  ro2  =  .492)  still  there  is  only  a  small  advantage 
in  accuracy  of  forecasting  when  we  consider  .r,,  a  func- 
tion of  the  two  variables  .Ti,  To  instead  of  a  simple  linear 
function  of  Xi.  If  we  were  to  regard  x\  =  (f)(xi,  x-i,  x^, 
...X,)  and  the  correlations  between  To  and  T3; .  .  .  ; 
0:0  and  x„  were  small,  little  or  nothing  would  be  gained 
in  accuracy  of  the  forecast  by  considering  these  addi- 
tional variables.^ 

The  method  which  we  have  adopted  in  case  of  three 
variables  is  general  in  its  character  and  may  be  ap- 
plied to  any  number  of  variables.  When  To  =  ^(ti,  Xo, 
T3, . .  .  x,)  and  the  variables  are  all  percentage  changes, 
it  is  possible  not  only  to  deal  with  the  dynamic  law  of 
demand  in  all  of  its  natural  complexity,  but  also  to  as- 
certain the  static  law  of  demand  giving  the  relation 
between  To  and  Ti  when  all  of  the  other  \'ariables  are 
equal  to  zero. 

The  problem  of  ascertaining  the  statistical  form  of 
the  law  of  demand  receives  by  this  method  an  adequate 
solution. 

1  "If  A  in  part  determines  B,  when  we  disregard  other  factors,  and 
C  in  part  determines  B,  when  we  disregard  all  else,  and  similarly  D  and 
E,  it  is  argued  that  all  these  part-determinations  can  be  added  together 
and  the  sum  will  finally  determine  B.  But  the  error  made  lies  in  the 
supposition  that  A,  C,  D,  E,  etc.,  are  themselves  independenl.  In  the 
imiverse  as  we  know  it,  all  these  factors  are  themselves  to  a  greater  or 
less  extent  associated  or  coi*related,  and  in  actual  experience,  but  little 
effect  is  produced  in  lessening  the  variability  of  B,  by  introducing  addi- 
tional factors  after  we  have  taken  the  first  few  most  highly  associated 
phenomena."     Pearson:  Grannnar  of  Science,  3rd  edition,  p.  172. 


CHAPTER  VI 

CONCLUSIONS 

The  business  of  economic  science,  as  distinguished  from  economic 
practice,  is  to  discover  the  routine  in  economic  affairs.  It  aims  to 
separate  out  the  elements  of  the  routine,  to  ascertain  their  interrela- 
tions, and  to  use  the  knowledge  of  their  connections  to  anticipate  ex- 
perience by  forecasting  from  known  changes  the  probabilities  of  corre- 
lated changes.  The  seal  of  the  true  science  is  the  confirmation  of  the 
forecasts;  its  -value  is  measured  by  the  control  it  enables  us  to  exercise 
over  ourselves  and  our  environment. 

Economists  theoretical  and  practical  have  grown 
impatient  with  anj^  form  of  speculation  that  is  not  of 
immediate  use.  The  present  generation  of  theoretical 
economists  expects  an  inquiry  to  be  dynamic,  to  take 
account  of  the  economic  flux,  to  show  a  routine  in 
change;  otherwise,  it  is  hypothetical,  static,  without 
significance  in  the  affairs  of  daily  life.  The  man  of 
affairs  must  be  con\inced  that  an  economic  inquiry 
will  either  make  directly  for  the  common  weal  or  else 
will  reveal  to  him,  in  the  pursuits  of  his  daily  life,  a 
source  of  individual  profit;  otherwise,  as  far  as  he  is 
concerned,  the  inquiry  is  academic,  visionary,  doc- 
trinaire. The  progress  of  the  new  type  of  economic 
theory  is  insured  by  the  fact  that  it  is  profitable  for 
practical  men  to  give  it  their  support. 

Forecasting  is  the  essential  aim  of  both  the  economic 
scientist  and  the  man  of  affairs.  According  to  the  most 
approved  doctrine,  economic  profit  has  its  origin  in 
economic  changes.  Other  forms  of  income  —  interest, 
wages,  and  rent  —  would  exist  in  a  ])iu'el3'  stationary 


1G4     Forecasting  the  Yietd  and  the  Price  of  Cotton 

state,  but  there  would  be  no  profit.  The  talent  of  the 
director  of  industry  in  the  modern  state  consists  in  his 
capacity  to  foresee  and  to  exploit  economic  changes, 
and  his  profit  is  proportionate  to  the  accuracy  with 
which  his  forecasts  are  made.  The  economic  scientist 
is  likewise  concerned  with  changes.  His  talent  con- 
sists in  his  capacity  to  separate  the  general  from  the 
accidental,  to  detect  the  routine  in  the  multitudinous 
details.  His  success  is  proportionate  to  the  simplicity 
and  generality  of  the  routine  that  he  may  disco\'er  and 
the  accuracy  with  which  he  is  able  to  foretell  the  size 
and  direction  of  future  changes. 

To  exemplify  the  simple  laws  of  economic  change, 
it  appeared  advisable  to  begin,  not  with  a  complex 
industrial  state  like  England  or  the  United  States, 
but  with  a  contemporary,  progressive  society  in  which 
the  whole  economic  life  is  dependent  upon  a  few  funda- 
mental interests.  It  seemed  that  no  territory  would 
afford  a  more  promising  field  for  such  a  quest  than  the 
Cotton  Belt  of  the  United  States.  Throughout  a  long 
period  it  has  been  recognized  that  in  the  vast  area  of 
the  Cotton  Belt  which,  with  Russia  excepted,  equals  in 
area  a  third  of  Europe,  ^^  Cotton  is  King.'"  And  not  only 
is  cotton  the  leading  staple  of  the  South,  but  three- 
fourths  of  the  world's  production  of  this  indispensable 
commodity  is  the  yield  of  our  Cotton  Belt.  Not  only 
does  the  change  in  the  price  and  yield  of  this  commodity 
affect  the  local  Cotton  Belt,  but,  to  the  extent  that 
cotton  enters  into  international  trade,  its  vicissitudes 
are  reflected  throughout  the  world. 

Would  it  be  possible  to  discover  the  routine  in  the 


Conclusions  165 

yield  and  price  of  cotton  so  that  the  knowledge  might 
be  used  for  purposes  of  forecasting? 

For  their  information  as  to  the  condition  and  promise 
of  the  growing  cotton  crop,  farmers,  brokers,  manu- 
facturers, and  merchants  rely  primarily  upon  the  re- 
ports of  the  United  States  Department  of  Agriculture. 
To  meet  the  public  demand,  the  Department  of  Agri- 
culture has  instituted  a  wonderful  statistical  organiza- 
tion. By  a  connection  with  many  thousands  of  cor- 
respondents, by  field-agents,  by  special  experts  in  crop 
estimates,  by  a  Bureau  of  Statistics  and  a  Crop-Report- 
ing Board,  information  has  been  systematically  gath- 
ered and  tabulated,  and  for  several  decades  monthly 
reports  ha^'e  been  issued  throughout  the  growth  season 
of  the  crop.  Extraordinary  precautions  have  been 
taken  to  pre\'ent  any  leakage  of  the  precious  informa- 
tion before  it  is  given  to  the  public. 

What  is  the  value  of  these  reports?  Since  they  are 
issued  under  the  aegis  of  the  Government  they  are 
assumed  to  be  fairl}^  accurate  representations  of  the 
facts,  and  official  authorities  have,  very  naturally, 
lost  no  opportunity  to  point  out  the  direct  advantage 
to  farmers  of  the  expenditure  of  pubUc  funds  for  this 
particular  purpose.  Speculators  have  regarded  the 
official  documents  as  of  value  for  their  ends,  and  nu- 
merous rumors  have  circulated  of  bribes  offered  and 
bribes  taken  for  advanced  information  as  to  the  con- 
tents of  the  reports.  But  what  is  the  value  of  these 
crop  reports  in  the  sense  of  their  degree  of  accuracy  as 
descriptions  of  actual  facts  and  their  measure  of  re- 
liability as  forecasts? 

Five  reports  are  issued  during  the  growth  season  of 


166     Forecasting  the  Yield  and  the  Price  of  Cotton 

cotton  and  refer  to  the  condition  of  the  crop  at  the  end 
of  May,  June,  July,  August,  and  September.  An  ex- 
amination of  these  reports  for  a  period  extending  over  a 
quarter  of  a  century,  1890-1914,  shows: 

(1)  That  the  May  report,  covering  the  condition  of 
the  cotton  crop  in  the  whole  country  at  the  end  of  May, 
is  so  erroneous  that  any  forecast  from  it  is  spurious. 
Any  money  that  changes  hands  as  a  result  of  the  report 
is  the  gain  or  loss  of  a  simple  gamble ; 

(2)  That  the  June  report  as  a  basis  of  forecasts  is 
better  than  the  May  report,  but  that  its  value  for  pur- 
poses of  forecasting  the  yield  per  acre  of  cotton  is 
negligible ; 

(3)  That  the  remaining  three  reports  —  for  July, 
August,  and  September  —  have  real  value,  but  the 
measurement  of  their  degree  of  accuracy  reveals  the 
anomaly  of  the  July  report  being  as  good  as  the  re- 
port for  August ; 

(4)  That  the  official  method  of  forecasting  favors 
the  farmers  by  giA'ing  an  underestimate  of  the  probable 
yield  of  cotton. 

These  are  the  concrete  facts  upon  which  the  practical 
man  in  touch  with  actual  affairs  bases  his  economic 
conduct.  Is  it  the  part  of  a  visionary  to  expect  to 
obtain  equally  reliable  forecasts  of  the  cotton  yield 
from  the  simple  reports  of  the  weather? 

Lord  Kelvin  has  told  us  that  "when  you  can  meas- 
ure what  you  are  speaking  about  and  express  it  in 
numbers,  you  know  something  about  it,  but  when  you 
cannot  measure  it,  when  you  cannot  express  it  in  num- 
bers, your  knowledge  is  of  a  meagre  and  unsatisfactory 
kind."     By  the  help  of  statistical  methods  that  rest 


Conclusions  167 

upon  the  theory  of  probabihty,  it  is  possible  to  measure 
the  precise  degree  of  accuracy  of  any  method  of  fore- 
casting, and,  consequently,  it  is  possible  to  compare  the 
relative  accuracy  of  forecasts  based  upon  official  re- 
ports and  forecasts  that  are  derived  from  the  records 
of  accumulated  rainfall  and  temperature  in  the  states 
of  the  Cotton  Belt.  For  purposes  of  comparison  we 
have  taken  the  four  leading  cotton  states,  which  to- 
gether produce  65  per  cent  of  the  entire  crop.  These 
four  states  are  Texas,  Georgia,  Alabama,  and  South 
Carolina.  Not  only  do  these  four  states  produce  the 
greater  part  of  the  total  cotton  crop,  but  they  represent 
the  weather  conditions  throughout  the  whole  Cotton 
Belt:  Texas  exemphfies  the  conditions  in  the  extreme 
Southwest;  Georgia  and  South  Carolina,  those  at  the 
other  extreme  on  the  Atlantic  Coast;  and  Alabama 
typifies  the  conditions  on  the  Gulf  of  Mexico.  We  shall 
consider,  for  these  representative  states,  the  results 
of  comparing  the  forecasts  from  the  condition  of  the 
growing  crop,  by  the  official  method;  and  the  forecasts 
from  the  changes  in  rainfall  and  temperature,  by  a 
method  which  we  have  fully  described. 

The  comparison  of  methods  will  be  made  clear  by 
examining  first  the  results  for  the  single  state  of  Georgia, 
From  calculations  based  upon  data  covering  a  quarter 
of  a  century,  we  find  in  case  of  Cieorgia: 

(1)  That  for  each  of  the  five  months  of  the  growth 
season  the  forecast  of  the  yield  per  acre  of  cotton  which 
is  based  upon  the  weather  data  is  decidedly  better 
than  the  forecast  from  the  condition  of  the  crop,  by 
means  of  the  official  formula; 

(2)  That  for  every  month  the  forecasts  from  the 


168     Forecasting  the  Yield  and  the  Price  of  Cotton 

weather  are  better  than  the  forecasts  a  month  later, 
by  the  official  method;  or,  more  definitely,  the  forecasts 
from  the  accumulated  weather  at  the  end  of  May, 
June,  July,  and  August  are  better  than  the  forecasts 
by  the  official  method  at  the  end  of  June,  July,  August, 
and  September; 

(3)  That  when  regard  is  paid  to  the  probable  errors 
of  the  coefficients  measuring  the  accuracy  of  the  fore- 
casts, then,  for  every  month,  the  forecasts  from  the 
weather  are  as  good  as  the  forecasts  two  months  later 
by  the  official  method.  Or,  more  definitely,  the  fore- 
cast from  the  May  weather  is  as  good  as  the  forecast 
by  the  official  method  at  the  end  of  July;  the  forecast 
from  the  joint  effect  of  the  May  and  June  weather  is 
as  good  as  the  forecast  by  the  official  method  at  the 
end  of  August,  and  the  forecast  from  the  accumulated 
weather  at  the  end  of  July  is  as  good  as  the  forecast 
by  the  official  method  at  the  end  of  September. 

We  shall  now  extend  our  comparison  to  the  results 
for  the  representative  states,  Texas,  Georgia,  Alabama, 
and  South  Carolina.  As  there  are  five  monthly  reports 
on  the  condition  of  the  growing  crop  and  we  have  taken 
four  representative  states,  there  are  twenty  cases  in 
which  the  forecasts  of  the  yield  per  acre  of  cotton  may 
be  compared: 

(1)  In  17  out  of  20  cases  the  forecasts  from  the 
weather  are  more  accurate  than  the  forecasts  from  the 
condition  of  the  crop,  by  the  official  method; 

(2)  For  all  of  the  representative  states  the  fore- 
casts by  the  official  method  from  the  May  condition 
of  the  crop  are  worthless.    By  contrast,  all  of  the  fore- 


Conclusions  169 

casts  from  the  May  weather  have  \'alue.  The  forecasts 
from  the  weather  for  Georgia  and  South  Carohna  are, 
at  the  end  of  May,  better  than  the  forecasts  by  the 
official  method  at  the  end  of  June,  and  about  as  good  as 
those  at  the  end  of  July ;  and  the  forecast  from  the  ^lay 
weather  in  Alabama  is  about  as  good  as  the  forecast 
by  the  official  method  at  the  end  of  September.  The 
value  of  the  forecast  from  the  May  weather  in  Texas 
is  negligible; 

(3)  For  three  out  of  the  four  representative  states 
the  forecasts  from  the  June  condition  of  the  crop,  by 
means  of  the  official  method,  are  worthless.  But  in 
all  three  cases  the  forecasts  from  the  accumulated 
weather  at  the  end  of  June  are  better  than  the  fore- 
casts by  the  official  method  at  the  end  of  July ; 

(4)  For  all  of  the  states  except  Texas  the  forecasts 
from  the  weather  give,  for  each  month,  more  accurate 
predictions  than  can  be  obtained  by  the  official  method 
from  the  condition  of  the  crop  one  month  later.  The 
forecasts  from  the  accumulated  weather  at  the  end  of 
Ma3%  June,  July,  and  August  are  better  than  the  fore- 
casts by  the  official  method  at  the  end  of  June,  July, 
August,  and  September; 

(5)  For  all  of  the  states  except  Texas  the  forecasts 
from  the  accumulated  weather  at  the  end  of  May,  June, 
and  July  are  about  as  good  as  can  be  obtained  b}^  the 
official  method  from  the  condition  of  the  crop  two  months 
later,  at  the  end,  respectively,  of  July,  August,  and 
September. 

As  the  routine  of  measurable  dependence  of  yield 
upon  the  weather  is  duo  to  the  presence  of  natural 


170     Forecasting  the  Yield  and  the  Price  of  Cotton 

causes,  it  might  easily  be  inferred  that  when  we  ino\-e 
to  strictly  social  facts,  no  such  routine  will  be  found. 
It  could  be  argued  that  the  price  of  cotton  results  from 
"the  law  of  suppl}^  and  demand";  the  supply  may  be 
predictable  because  it  is  primarily  dependent  upon 
natural  causes;  but  the  demand  is  a  social  fact  and  is 
the  resultant  of  many  indi\'idual  choices  each  of  which, 
in  its  turn,  is  dependent  upon  many  variable  factors. 
By  such  a  priori  reasoning  it  would  be  easy  to  conclude 
that  it  is  futile  to  attempt  to  find  a  predictable  routine 
in  the  dependence  of  the  price  of  cotton  upon  the  size 
of  the  crop. 

But  again  we  are  reminded  of  Lord  Kehin's  state- 
ment that ' '  when  you  can  measure  what  you  are  speak- 
ing about  and  express  it  in  numbers,  you  know  some- 
thing about  it,  but  when  you  cannot  measure  it,  when 
you  cannot  express  it  in  numbers,  your  knowledge  is  of 
a  meagre  and  unsatisfactory  kind."  Our  researches 
have  shown  that  there  is  a  dynamic  law  of  demand  — 
a  law  that  connects  the  price  of  cotton  with  the  size 
of  the  crop  —  and  that  the  knowledge  of  the  law  would 
have  made  possible  the  prediction  of  the  price  of  cotton 
from  1890  to  1914  with  a  degree  of  accuracy  higher 
than  that  attained  by  the  formula  of  the  Department 
of  Agriculture  in  the  annual  prediction  of  the  size  of  the 
crop  at  the  end  of  September.  Upon  the  appearance 
of  the  monthly  cotton  reports,  great  sums  of  money 
exchange  hands  because  of  the  light  they  are  supposed 
to  throw  upon  the  probable  size  of  the  crop.  The  most 
reliable  report  is,  of  course,  the  one  nearest  the  harvest, 
but  the  accuracy  of  the  forecasts  of  yield  that  are  based 
upon  this  report  is  less  than  the  accuracy  with  which 


Conclusions  171 

the  price  of  cotton  can  be  predicted  from  the  size  of  the 
crop,  by  means  of  the  law  of  demand. 

Both  the  yield  and  the  price  of  cotton,  therefore,  are 
so  much  a  matter  of  routine  that  they  admit  of  pre- 
diction with  a  high  degree  of  precision. 

In  laying  the  foundation  of  the  modern  type  of  Eco- 
nomic Theory,  Je\'ons  foresaw  the  necessity  of  the 
simultaneous  development  of  its  deductive  and  its 
inductive  phases: 

^'I  know  not  when  we  shall  have  a  perfect  system  of 
statistics,  but  the  want  of  it  is  the  only  insuperable 
obstacle  in  the  way  of  making  Economics  an  exact 
science.  In  the  absence  of  complete  statistics,  the 
science  will  not  be  less  mathematical,  though  it  will  be 
immensely  less  useful  than  if  it  were,  comparatively 
speaking,  exact.  A  correct  theory  is  the  first  step  to- 
wards improvement,  by  showing  what  we  need  and 
what  we  might  accomplish." 

"The  deductive  science  of  Economics  must  be  veri- 
fied and  rendered  useful  by  the  purely  empirical  science 
of  statistics.  Theory  must  be  invested  with  the  reality 
and  life  of  fact."  ^ 

In  the  opinion  of  Professor  Marshall  the  pressing 
need  of  economic  science  at  the  present  time  is  "the 
quantitative  determination  of  the  relative  strength  of 
different  economic  forces."  -    And  Professor  Pareto,  in 

'  Jevons:  Theory  of  Folitical  Econoinj/,  3rd  (Mlition,  pp.  12,  22.  Cf. 
Jovons:  Principles  of  >'^cienee,  chapter  xxii,  ciul  of  the  section  on  "Illus- 
tration of  Empirical  (Quantitative  Laws." 

-  Cf.  the  address  of  Professor  W.  J.  Ashley  as  President  of  Section  F 
of  the  British  Association  for  the  Aflvancement  of  Science.  Report  of 
the  lirilinh  Association  for  the  Adnutcenient  of  Science,  1907,  p.  o91. 


172     Forecasting  the  Yield  and  the  Price  of  Cotton 

like  spirit,  points  out  the  conditions  of  the  fui'thei'  de- 
velopment of  our  science : 

''The  progress  of  Political  Economy  in  the  future 
will  depend  in  great  part  upon  the  investigation  of 
empirical  laws,  derived  from  statistics,  which  will 
then  be  compared  with  known  theoretical  laws,  or  will 
suggest  the  derivation  from  them  of  new  laws."  ^ 

The  idea  that  I  should  like  to  emphasize  is  that 
because  of  the  recent  development  of  statistical  theory 
and  the  improvement  in  the  collection  of  statistical 
data,  we  are  now  able  to  meet  the  needs  so  clearly  de- 
scribed by  the  masters  of  the  science. 

The  great  advance  in  the  methodology  of  deductive 
economics,  after  Cournot's  epoch-making  work,  was 
initiated  by  Leon  Walras  in  his  use  of  simultaneous 
equations  for  the  purpose  of  completely  surveying  the 
interrelated  factors  in  the  problems  of  exchange,  pro- 
duction, and  distribution.  It  was  necessary  in  his  work 
and  in  the  work  of  his  successors  to  begin  with  a  simple, 
hypothetical  construction  and  to  approach  the  concrete 
problem  by  the  introduction  of  an  increasing  number 
of  complicating  factors.  The  equations  expressing 
the  relations  between  the  variables  were,  of  necessitj^, 
arbitrary,  but  the  device  made  possible  the  envisaging 
of  all  the  elements  in  the  problem,  and  suggested  the 
types  of  their  interrelations.  But  economic  theory 
has  now  reached  the  stage  where,  according  to  Professor 
Marshall,  there  is  need  of  a  "quantitative  determma- 
tion  of  the  relative  strength  of  the  different  economic 

^  Giornale  degli  Economisti,  Maggio,  1907,  p.  8G6.  "II  progresso 
deir  Economia  jjolitica  dipendera  pel  futuro  in  gran  ])arte  dalla  rioorca 
di  leggi  emj^iriche,  ricavate  dalla  statistica,  e  che  si  paragoneranno  poi 
colle  leggi  teoriche  note,  o  che  ne  faranno  conoscere  di  niiove." 


Conclusions  173 

forces";  and,  according  to  Professor  Pareto,  empirical 
laws  must  be  deri\-ed  from  statistics  for  the  double 
.purpose  of  comparing  them  with  known  theoretical 
laws  and  of  gaining  bases  for  new  theoretical  develop- 
ments. 

The  statistical  theory  of  multiple  correlation  is  per- 
fectly adapted  to  these  demands.  No  matter  what 
may  be  the  number  of  factors  in  the  economic  problem, 
it  is  specially  fitted  to  make  a  "quantitative  determi- 
nation" of  their  relative  strength;  and  no  matter  how 
complex  the  functional  relations  between  the  variables, 
it  can  derive  "empirical  laws"  which,  by  successive 
approximations,  will  describe  the  real  relations  with 
increasing  accurac}^  The  mathematical  method  of 
deductive  economics  gives  a  coup  d'ceil  of  the  factors 
in  the  problem;  the  statistical  method  of  multiple 
correlation  affixes  their  relative  \'alue  and  reveals  the 
laws  of  their  association.  The  mathematical  method 
begins  with  an  ultra-hypothetical  construction  and 
then,  by  successive  complications,  approaches  a  theoret- 
ical description  of  the  concrete  goal.  The  method  of 
multiple  correlation  reverses  the  process:  It  begins 
with  concrete  reality  in  all  of  its  natural  complexity 
and  proceeds  to  segregate  the  important  factors,  to 
measure  their  relative  strength,  and  to  ascertain  the 
laws  according  to  which  they  produce  their  joint  efTect. 
When  the  method  of  multiple  correlation  is  thus  applied 
to  economic  data  it  invests  the  findings  of  deductive 
economics  with  "the  reafity  and  life  of  fact";  it  is  the 
Statistical  Complement  of  Deducti\'e  Economics. 


T 


HE  following  pages  contain  advertisements  of  Mac- 
millan  books  by  the  same  author. 


Economic    Cycles:    Their    Law    and 
Cause 

By  henry  LUDWELL  MOORE 

Professor  of  Political  Economy  in  Columbia  University 

8vo,  S2.00 

Extract  from  the  Introduction:  ''There  is  a  consid- 
erable unanimity  of  opinion  among  experts  that,  from 
the  purely  economic  point  of  view,  the  most  general 
and  characteristic  phenomenon  of  a  changing  society  is 
the  ebb  and  flow  of  economic  life,  the  alternation  of 
energetic,  buoyant  acti\ity  with  a  spiritless,  depressed 
and  uncertain  drifting.  .  .  .  What  is  the  cause  of  this 
alternation  of  periods  of  activity  and  depression?  \Miat 
is  its  law?  These  are  the  fundamental  problems  of 
economic  dynamics  the  solution  of  which  is  offered  in 
this  Essay." 

Comments  of  Specialists 

Moore's  book  is  so  im])ortatit  that  it  is  sure  to  be  widely  criti- 
cized. .  .  .  Yet  so  far  as  the  fundamental  conclusions  are  concerned 
the  book  is  so  firmly  grounded  on  a  vast  l:)ody  of  facts  that  its  main 
line  of  argmiient  seems  unassailable.  .  .  .  Moore  has  gone  much 
further  than  his  predecessors  and  has  removed  his  subject  from  the 
realm  of  probability  to  that  of  almost  absolute^  certainty.  Here- 
after there  can  be  little  question  that  apart  from  such  influences  as 
the  depreciation  in  gold,  or  great  calamities  like  the  war,  the  general 
trend  of  economic  conditions  in  this  country  is  (-losely  depend(>nl 
upon  cj'clical  variations  in  the  weather."  —  Ellsworth  Huxting- 
TON,  in  the  Geographical  Review. 

In  reply  to  the  question:  "  \\'hat  are  the  two  best  books  you 
have  read  recently,"  President  Butler  named,  as  one  of  the  two 
books.  Professor  Moore's  Eeonnmic  Cycles  because  of  its  lieing  "an 


original  and  very  stiuiulatiuf!;  study  in  economic  theory  with  riuick 
applications  to  practical  business  affairs."  —  Nicholas  Murhay 
Butler,  in  the  New  York  World. 

"Professor  Moore  is  known  among  scliolars  as  one  of  the  keenest 
and  most  cautious  of  investigators.  .  .  .  His  novel  methods  of 
investigation  constitute  an  additional  claim  upon  our  interest;  the 
problem  of  the  crisis  has  never  yet  been  approached  in  precisely  this 
way."  —  Alvin  S.  Johnson,  in  the  Neio  Republic. 

"This  book  indicates  a  method  of  utilizing  (economic)  data  .  .  . 
that  is  worthy  of  the  highest  commendation."  —  Allen  Hazen,  in 
the  Engineering  Neios. 

"  If  the  promise  of  Professor  Moore's  convincing  Essay  is  fulfilled, 
economics  will  become  an  approximately  exact  science.  ...  If 
progress  is  made  in  the  direction  of  such  a  goal  as  a  result  of  this 
work,  it  will  be  the  economic  contribution  of  a  century,  and  will 
usher  in  a  new  scientific  epoch."  —  Roy  G.  Blakey,  in  the  Times 
Annalist. 

"The  agricultural  theor^v  of  cycles  has  found  a  new  and  brilliant 
exponent  in  Professor  Henry  L.  Moore."  —  Wesley'  Clair  Mit- 
chell, in  the  American  Yearbook. 

"If  his  methods  stand  the  test  of  experience,  and  can  l^e  widely 
adopted,  the  field  of  business  may  be  revolutionized  so  far  as  it  con- 
cerns the  enterpriser  because  the  measuring  of  the  force  of  under- 
lying, fundamental  conditions  will  become  approximately  accurate 
and  the  function  of  the  enterpriser  will  thereb,v  be  reduced." 
Magazine  published  by  Alexander  Hamilton  Institute. 

"L'auteur  a  mis  a  son  service  des  procedes  mathematiques  et 
statistiques  raffines  et  elegants  .  .  .  celui-ci  a  ecrit  un  livre  bril- 
lant."  —  Umberto  Ricci,  in  Scientia. 


THE    MACMILLAN    COMPANY 

Publishers         64-66  Fifth  Avenue         New  York 


Laws  of  Wages 

An  Essay  In  Statistical  Economics 

By  henry  LUDWELL  MOORE 

Professor  of  PolUical  Economy  in  Columbia  University 

8vo,  SI. 60 
Extract  from  the  Introduction:  ''In  the  following 
chapters  I  have  endeavored  to  use  the  newer  statistical 
methods  and  the  more  recent  economic  theory  to  ex- 
tract, from  data  relating  to  wages,  either  new  truth 
or  else  truth  in  such  new  form  as  will  admit  of  its  being 
brought  into  fruitful  relations  with  the  generalizations 
of  economic  science." 

Comments  of  Specialists 

"Professor  IMooi-e  l)riiigs  to  his  task  a  wide  acquaintance  with 
the  most  difficult  parts  of  the  Hterature  of  economics  and  statistics, 
a  full  appreciation  of  its  large  jM-oblems,  a  judicial  spirit  and  a  dig- 
nified stjde."  —  F.  W.  Taussig,  in  the  Quarterly  Joiinidl  of  Eco- 
nomicH. 

"Statistics  of  the  ordinary  official  kind  have  often  served  to 
support  the  arguments  of  political  economists.  But  this  is  the  first 
time,  we  believe,  that  the  higher  statistics,  which  are  founded  on  the 
Calculus  of  Probabilities,  have  been  used  on  a  large  scale  as  a  but- 
tress of  economic  theoiy."  — F.  Y.  Edgeworth,  in  the  Economic 
Journal. 

"Professor  ^loore  has  broken  new  ground  in  a  most  interesting 
field,  and  while  we  maj'  differ  from  him  in  the  weight  to  be  attached 
to  this  or  that  r(>sult  or  the  inlcr])r('tation  to  be  placed  on  some 


observed  cocfficieut,  wo  uui\'  offer  cord'uil  coiifi,r:ilulalious  on  the 
work  as  a  whole."  —  G.  Y.  Yule,  in  the  Journal  of  the  Royal 
Slatidical  Sociefi/. 

"Die  Fru('htl)Mrkeit  fler  verwciideteii  Mcthode  schciul  mil'  (hirch 
(Uese  Untersuchungen  zweifellos  erwieseii,  el)(>nso  wie  die  Erreich- 
barkeit  des  Ziels,  (he  Theorie  gauz  dicht  an  (He  Zahlenaus(h'iicke  der 
wirtschaftlichen  Tatsachen  hcranziibringen.  L'lul  das  ist  eine  Tat, 
zu  dcr  der  Alitor  uur  zu  l)eghick\vuuschen  ist.  .  .  .  Hat  (his  Buch 
auch  auf  der  Hand  liegende  Fehler  —  in  der  Zukunft  wird  mau  sich 
seiner  als  der  ersten  klaren,  einfachen  iind  zielbewussten  Darlegung 
und  Exemplifizierung  der  Anwendung  der  'hoheren  Statistik'  auf 
okonomische  Prol)leine  dankljur  erinnern."  —  Joseph  Schumpeter, 
in  the  Archiv  fiir  Sozialwissenschaft  und  SozialpoJilik. 

"Non  seulenient  il  nous  enseigne  I'emploi  d'unc  m^'thode  qui 
dans  de  certaines  Hmites  pent  etre  tres  feconde.  Mais  encore  son 
habilete  personnelle  dans  le  maniement  de  cette  methode  est  tres 
rcelle.  II  sait  scruter  les  statistiques  d'une  fa(,'on  fort  p?netrante  et 
exposer  les  resultats  de  ses  recherches  avec  beaucoup  d'elegance. 
Le  lecteur  frangais  en  particulier,  appreciera  I'ingeniosits  avec 
laquelle  il  tire  des  statistiques  frangaises  des  inductions  souvent 
nouvelles  et  justes."  —  Albert  Aftalio.v,  in  the  Revue  d'hidoire 
des  doctrines  economiques. 

"Alcuni  dei  risultati  ottenuti  dall'autore,  sono  nuovi  e  suggestivi 
e  da  essi  molto  conclusioni  si  possono  trarre  (cui  I'autore  accenna 
nel  capitolo  finale  della  sua  opera)  sia  rispetto  alle  teorie  del  salario 
("he  rispetto  alia  politica  sociale.  II  libro  e  insonnna,  ripetianio,  un 
contributo  molto  importante  aU'investigazione  scientifica  dei  fe- 
nomeni  economici  e  vorremmo  che  esso  stimolasse  parecchi  altri 
studiosi  a  fare  per  altre  industrie  o  per  altri  paesi,  recerche  analoghe.'' 
—  Constantino  Bresciani  Turroxi,  in  the  Giornnle  degli  Econo- 
inisti. 

THE    MACMILLAN    COMPANY 

Publishers  64-66  Fifth  Avenue  New  York 


UNIVERSITY  OF  CALIFORNIA  AT  LOS  ANGELES 

THE  UNIVERSITY  LIBRARY 
This  book  is  DUE  on  the  last  date  stamped  below 


SEP  1  0  1952 


iSM  (fe,  ii.  '--^^ji 


Form  L-9 
JOm-12, '39(3386) 


OWTERSITY  OF  CAUtFORNU 

AT 

LOS  ANG£L£S 

UBRARY 


UCLA-Young   Research    Library 

HD9074   .M78f 

y 


III  II II  II  II 
L  009  568   750   5 


HD 
9074 
M78f 


SUPPLIED     BY 

THE   SEVEN    BOOKHUNTERS 

STATION   0,   BOX  22  —  NEW   YORK   CITY 
Out-of-PrInt   Books 


